LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Class 


. 


AN 


INTRODUCTION 


TO 


NATURAL  PHILOSOPHY; 


DESIGNED    AS    A 


TEXT-BOOK 

FOB    THE    USE    OP 

STUDENTS     IN     COLLEGE. 
BY  DENISON  OLMSTED,   LL.D., 

PROFESSOR    OF    NATURAL    PHILOSOPHY    AND    ASTRONOMY    IN    YALE    COLLEGE, 

AND  E.   S.   SNELL,    LL.D., 

PROFESSOR     OF     MATHEMATICS     AND     NATURAL     PHILOSOPHY     IN     AMHERST    COLLBGK 

THIRD    REVISED    EDITION. 

BY  RODNEY  G.   KIMBALL,   A.M., 

PROFESSOR   OF  APPLIED   MATHEMATICS   IN   BROOKLYN   POLYTECHNIC  TOSTITU*  f 


NEW    YORK: 
COLLINS    &   BROTHKR 

414   BROADWAY. 


OU 


Entered  according  to  Act  of  Congress,  in  the  year  1844,  by 

DENISON  OLMSTED, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


REVISED  EDITION. 
E.itered  according  to  Act  of  Congress,  in  the  year  1860,  by 

JULIA    M.    OLMSTED, 

FOB  THE  CHILDREN  OF  DENISON  OLMSTED,  DECEASED, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Connecticut. 


SECOND  REVISED  EDITION. 
Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

JULIA  M.  OLMSTED, 

FOB  THE  CHILDBEN  OP  DBNISON  OLMSTED,  DECEASED, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


.  THIBD  REVISED   EDITION. 

Copyright,  1882, 

By   JULIA   M.    OLMSTED, 

FOB  THE  CHILDBEN  OF  DENISON  OLMSTED,  DECEASED. 


PREFACE. 


present  revision  of  Olmsted's  Natural  Philosophy  has 
been  made  in  accordance  with  the  plan  of  the  former 
editions,  the  object  being  to  produce  a  text-book  in  physics 
adapted  to  the  requirements  of  a  college  course ;  it  is  believed 
that  the  work  as  now  presented  includes  all  that  is  necessary  in 
this  department  of  a  liberal  education. 

While  students  in  technical  schools  may  need  special  treatises 
upon  various  subdivisions  of  physics,  the  general  course  here 
offered  will  prove  a  sufficient  preparation  for  such  technical  work. 

The  rapid  growth  of  science  has  made  it  necessary  to  add 
materially  to  the  size  of  the  volume,  although  many  facts  of 
interest  and  descriptions  of  practical  applications  have  been  neces- 
sarily omitted  that  the  subjects  treated  of  might  be  presented 
within  reasonable  limits.  The  aim  of  the  editor  has  been  to 
present  the  principles  of  physics,  and  not  to  compile  an  exhaustive 
treatise.  For  this  reason,  also,  only  such  new  engravings  have  been 
added  as  were  necessary  to  make  clear  the  meaning  of  the  text 
and  to  impress  the  facts  upon  the  memory ;  the  professor  in  charge 
of  the  department  is  thus  left  free  to  pursue  such  course  of  experi- 
mental illustration  as  his  own  experience  and  the  facilities  at  his 
command  render  most  desirable. 


JV  PREFACE. 

The  Appendix  (Applications  of  the  Calculus)  having  found 
favor  with  many,  is  retained  unchanged. 

Part  L,  Mechanics,  has  been  but  slightly  altered,  and  there- 
fore the  indebtedness  to  Professor  Joseph  Ficklin,  expressed  by 
the  author  of  the  revision  of  1870,  is  properly  acknowledged 
here. 

BROOKLYN,  N.  Y.,  August,  1882. 


CONTENTS. 


USTTKODUCTION. 

PAGES 

Classification  of  the  Physical  Sciences. — Definitions  relating  to  Matter. — 
Properties  of  Matter.— Branches  of  Natural  Philosophy  ...        1-3 


PART    I.— MECHANICS. 

CHAPTER    I. 

Motion  Classified. — Uniform  Motion. — Momentum. — Forces  classified. — 
Measure  of  Force. — Laws  of  Motion. — Gravity. — Its  Laws  .  .  4-12 

CHAPTER    II. 

Motion  produced  by  Constant  Forces. — Relations  of  Space,  Time,  and 
Acquired  Velocity. — Uniform  and  uniformly  varied  Motion  combined. 
— Application  of  the  Laws  to  the  Fall  of  Bodies. — Determination  of  the 
Acceleration  due  to  a  given  Force.  —  Atwood's  Machine. — Work. — 
Living  Force.— Applications  of  Living  Force  to  Work  .  .  .  13-35 

CHAPTER    III. 

Motion  produced  by  Two  or  more  Forces. — Parallelogram  of  Forces. — 
Triangle  of  Forces. — Polygon  of  Forces. — Curvilinear  Motion. — Calcu- 
lation of  the  Resultant. — Resolution  of  Motion. — Resolution  of  a  Force 
to  find  its  Efficiency  in  a  given  Direction. — Resultant  found  by  Rect- 
angular Axes. — Analytical  Expression  for  Resultant. — Principle  of 
Moments. — Parallel  Forces. — Couples. — Parallelepiped  of  Forces. — 
Three  Rectangular  Axes. — Equilibrium  of  Forces  in  different  Planes. — 
Forces  resisted  by  a  smooth  Plane  .......  25-43 

CHAPTER    IV. 

Centre  of  Gravity. — Centre  of  Gravity  of  Two  Equal  Bodies. — Of  Two 
Unequal  Bodies. — Equal  Moments. — Three  or  more  Bodies. — Triangle. 
Polygons. — Perimeters. — Pyramid. — Centre  of  Gravity  referred  to  a 
Point,  or  a  Plane. — Trapezoid. — Centrobaric  Mensuration. — Different 
Kinds  of  Equilibrium. — Motion  of  the  Centre  of  Gravity  of  a  System. — 
Mutual  Action  among  the  Bodies  of  a  System  .....  43-58 


yj  CONTENTS. 

CHAPTER    V. 

PAGES 

Collision  of  Inelastic  Bodies.—  Collision  of  Elastic  Bodies.—  Series  of 
Bodies,  Equal,  Increasing,  Decreasing.  —  Living  Force  lost  in  the  Col- 
lision of  Inelastic  Bodies.  —  Preserved  in  the  Collision  of  Elastic 
Bodies.  —  Impact  on  a  Plane  ...  .  .  .  .  .  59-64 

CHAPTER    VI. 

Classification  of  Machines.  —  Levers.  —  Equal  Moments.  —  Acting  Dis- 
tance. —  Compound  Lever.  —  Balance.  —  Steelyard.—  Platform  Scales.  — 
Wheel  and  Axle.  —  Differential  Pulley.  —  Compound  Wheel  and  Axle.  — 
Connection  by  Teeth  and  by  Bands.  Pulley.  —  Fixed,  Movable,  Com- 
pound. —  Rope  Machine.  —  Branching  Rope.  —  Inclined  Plane.  —  Relations 
of  Power,  Weight,  and  Pressure.  —  Bodies  Between  Two  Planes.  —  On 
Two  Planes.  —  Screw.  —  Combined  with  the  Lever.  —  Endless  Screw.  — 
Wedge.  —  Knee  Joint.  —  Virtual  Velocities.  —  Friction  in  Machinery.  — 
Sliding  Friction.  —  Friction  of  Axes.  —  Rolling  Friction.  —  Limiting 
Angle  of  Friction  ...  .  .  .  r  .  '.  .  .  65-96 


CHAPTER    VII. 

Motion  on  Inclined  Planes.  —  Formulas.—  Descent  on  the  Chords  of  a 
Circle.  —  Loss  in  passing  from  one  Plane  to  another.  —  No  Loss  on  a 
Curve.  —  Similar  Systems  of  Planes  and  Curves.  —  Pendulum.  —  Calcu- 
lation of  the  Length.  —  Point  of  Suspension  and  Centre  of  Oscillation 
interchangeable.  —  Calculation  of  the  Time  of  Oscillation.  —  Relation  of 
Gravity  and  Length.  —  Compensation  Pendulum  ....  96-108 


CHAPTER    VIII. 

Central  Forces.  —  Centrifugal  Force  in  Circular  Motion.  —  Two  Bodies 
revolving  about  their  Centre  of  Gravity.  —  Centrifugal  Force  on  the 
Earth.—  Composition  of  two  Rotary  Motions.—  The  Gyroscope  .  108-113 


PART    II.— HYDROSTATICS. 

CHAPTER    I. 

Liquids  distinguished  from  Solids  and  Gases. — Transmitted  Pressure. — 
Hydraulic  Press.— Equilibrium  of  a  Liquid. — Curvature  of  a  Liquid 
Surface. — Pressure  as  Depth. — Hydrostatic  Paradox. — Amount  of  Pres- 
sure.— Determination  of  Thickness  of  Cylinder  to  resist  given  Pres- 
sure.— Same  Level  in  connected  Vessels. — Artesian  Wells. — Centre  of 
Pressure.— Loss  of  Weight  in  Water.— Floating  Bodies.— Metacentre.— 
Specific  Gravity.— Methods  of  finding  Specific  Gravity.— Table  of 
Specific  Gravities. — Cohesion  and  Adhesion. — Capillary  Action. — Effects 
of  Capillarity  on  floating  Bodies 114-137 


CONTENTS.  vii 

CHAPTEK    II. 

PAGES 

Liquids  in  Motion. — Depth  and  Velocity  of  Discharge. — Descent  of  Sur- 
face.— Orifices  in  Different  Situations. — Vena  Contracta. — Friction  in 
Pipes.  —  Jets. — Rivers. — Lifting  Pumps. — Chain  Pump. — Centrifugal 
Pump.— Hydraulic  Ram.— Water  Wheels.— Turbine.— Barker's  Mill- 
Resistance  to  Motion  in  a  Liquid. — Water  Waves. — Molecular  Move- 
ments.—Sea  Waves  137-151 


PART    III.— PNEUMATICS. 


CHAPTER    I. 

Gases  distinguished  from  Liquids. — Tension  of  Gases. —Diffusion. — Os- 
mose.— Weight  and  Pressure. — Buoyancy. — Torricelli's  Experiment. — 
Pressure  of  Air  measured. — Pascal's  Experiment. — Mariotte's  Law.— 
Dalton's  Law. — Laws  of  Mixture  of  Gases  and  Liquids. — Barometers. — 
Pressure  and  Latitude.  —  Diurnal  Variation.  —  Barometer  and  the 
Weather. — Heights  measured  by  the  Barometer  .  .  .  153-164 

CHAPTER    II. 

The  Air-Pump. — Sprengel's  Pump. — Air  Condenser. — Bellows.— Siphon. 
— Siphon  Fountain. — Suction  Pump. — Calculation  of  the  Force  re- 
quired.— Force  Pump. — Fire  Engine. — Hero's  Fountain. — Manometers. 
— Apparatus  for  preserving  a  constant  Level  in  a  Reservoir  .  164-175 

CHAPTER    III. 

Virtual    Height    of    the   Atmosphere.  —  Changes    in    Density. — Actual 
Height. — Trade  Winds. — Land  and  Sea  Breezes. — Current  in  a  Medium. 
-Ventilators. — A  Stream  meeting  a  Surface. — Vortices     .        .        175-181 


PART    IV.— ACOUSTICS. 


CHAPTER    I. 

Vibrations  the  Cause  of  Sound.— Sonorous  Bodies.— Air  as  a  Medium.— 
Velocity  of  Sound  in  Air. — Diffusion  of  Sound. — Nature  of  Acoustic 
Waves.— Intensity  of  Sound.— Gases,  Liquids,  and  Solids  as  Media.— 
Mixed  Media  ' 183-191 

CHAPTER    II. 

Reflection  of  Sound. — Echoes.  —  Concentrated  Echoes. — Resonance  of 
Rooms.— Halls  for  Public  Speaking.— Refraction.— Inflection  .  196-197 


viii  CONTENTS. 

CHAPTEK    III 


PAGES 


Vibrations  in  Musical  Sounds. — Pitch.— Monochord. — Time  of  one  Vibra- 
tion.— Vibration  of  a  String  in  Parts. — Longitudinal  Vibrations  of 
Strings. — Vibrations  of  a  Column  of  Air. — Position  of  Nodes  deter- 
mined experimentally. — Modes  of  exciting  Vibrations  in  Pipes. — 
Vibrations  of  Rods  and  Laminae.— Of  Wires.— Of  Chladni's  Plates.— Of 
Bells.— The  Voice.— The  Ear  .  .  .  .  .  .  .  197-211 

CHAPTEE    IV. 

Numerical  Relations  of  Musical  Notes. — The  Scale.— Chords  and  Dis- 
cords.— Temperament. — Harmonics. — Overtones. — Quality  of  Tone. — 
Communication  of  Vibrations. — Crispations. — Interference. — Number 
and  Length  of  Waves  for  each  Note. — Doppler's  Principle. — Vibrations 
visibly  projected.— Telephone  .  .  .  .  v  *  .  211-223 


PAET    V.— OPTICS. 

CHAPTER    I. 

Definitions  and  First  Principles. — Velocity  of  Light. — Law  of  Intensity. — 
Brightness  the  same  at  all  Distances. — Absorption. — Photometers. — 
Shadows  .  .  .  .  .  .  .  .  .  .  .  224-230 

CHAPTER    II. 

Reflection. — From  Plane  Mirror. — Effect  of  Spherical  Mirrors.— Conju- 
gate Foci. — Images  by  Plane  Mirror. — Object  and  Image  symmetri- 
cal.— Requisite  Length  of  Mirror. — The  Sextant. — Multiplied  Images 
by  Two  Mirrors,  either  Parallel  or  Inclined. — The  Kaleidoscope. — 
Images  by  Spherical  Mirrors. — Caustics. — Spherical  Aberration  .  230-245 

CHAPTER    III. 

Refraction. — Law  of  Refraction. — Limit  of  Transmission  from  denser 
Medium.  —  Opacity  of  mixed  transparent  Media.  —  Transmission 
through  parallel  Plane  Surfaces. — Determination  of  relative  Indices  of 
Refraction. — Transmission  through  a  Prism. — Light  through  One  Sur- 
face, Plane,  Convex,  or  Concave. — Lenses. — Effect  of  Convex  and  Con- 
cave Lens. — Optic  Centre. — Conjugate  Foci. — Powers  of  Lenses  practi- 
cally determined. — Equivalent  Combinations. — Images  by  Lenses. — 
Caustics.— Spherical  Aberration.— Atmospheric  Refraction.— Mirage  245-262 

CHAPTER    IV. 

The  Prismatic  Spectrum. — Colors  recombined. — Complementary  Colors. — 
Natural  Color  of  Bodies. — Frauenhofer's  Lines. — Characteristic  Lines  of 
the  Elements.  —  Composition  of  the  Sun's  Surface.  —  Dispersion. — 
Chromatic  Aberration  of  Lenses.— Achromatism.— Irrationality  of  Dis- 
persion .  .  .  263-270 


C  O  N  T  E  N  T  S  .  ix 

CHAPTER    V. 

PAGES 

The  Rainbow. — Primary  Bow. — Course  of  Rays  in  the  Secondary  Bow. — 
Axis  of  the  Bow. — Its  circular  Form. — Colors  reversed  in  the  Two 
Bows. — Colored  Borders  of  illuminated  Segments  of  the  Sky. — Halo.— 
Mock  Sun  .  •  270-277 

CHAPTER    VI. 

The  Wave  Theory. — Nature  of  the  Wave. — Reflection  and  Refraction 
according  to  the  Wave  Theory. — Relation  of  Angles  of  Incidence,  Re- 
flection, and  Refraction.  —  Interference.  —  Striated  Surfaces.  —  Thin 
Laminae. — Thickness  of  Lamina  in  Newton's  Rings. — Diffraction  or 
Inflection. — Light  through  small  Apertures. — Length  and  Number 
of  Luminous  Waves — Calorescence  and  Fluorescence. — Phosphores- 
cence .  .  i 277-288 

CHAPTER    VII. 

Change  of  Vibrations  in  Polarized  Light. — Polarizing  by  Reflection. — 
The  Polarizing  Angle. — Polarization  by  a  Bundle  of  Plates. — Polariza- 
tion by  Absorption. — Double  Refraction. — Polarizing  by  doubly  refract- 
ing Crystals. — Different  Kinds  of  Polarization. — Nicol's  Prism. — Color 
by  Polarized  Light.— Colored  Rings  .  ...  .  ....  288-296 

CHAPTER    VIII. 

Images  by  Light  through  small  Apertures. — Effect  of  Convex  Lens. — 
The  Eye. — Vision. — Adaptations  of  the  Eye. — Long-sightedness. — The 
Blind  Point. — Continuance  of  Impressions. — Subjective  Colors. — Irra- 
diation.— Magnitude  and  Distance  associated. — Binocular  Vision  296-304 

CHAPTER    IX. 

The  Camera  Lucida. — Microscopes. — The  Magnifying  Power. — Magic 
Lantern. — The  Astronomical  Telescope. — Powers  of  the*Telescope. — 
Terrestrial  Telescope. — Galileo's  Telescope. — Gregorian  Telescope. — 
Cassegrainian  Telescope. — Newtonian  Telescope. — Herschelian  Tele- 
scope.—Eye-pieces  304-314 


PART    VI.— HEAT. 

CHAPTER     I. 

Nature  of  Heat. — Expansion. — Coefficients  of  Expansion. — Strength  of 
the  Thermal  Force. — Expansion  of  Liquids. — Exceptional  Case. — Ex- 
pansion of  Gases. — Thermometers. — Systems  of  Graduation. — Reduc- 
tion from  one  Scale  to  another. — Absolute  Zero  of  Temperature  .  315-321 


x  CONTENTS. 

CHAPTER    II. 

PAGES 

Modes  of  communicating  Heat. — Conduction. — Effects  of  Molecular 
Arrangement.  — Differences  in  Conductivity. — Convection. — Determina- 
tion of  Temperature  of  Maximum  Density  of  Water. — Radiation. — 
Equalization  of  Temperature  by  Radiation. — Reflection. — Absorption. — 
Diathermancy ,  -  ••-.  .  .  .  321-328 

CHAPTER    III. 

Specific  Heat. — Methods  of  Finding. — Apparent  Conduction  affected  by 
Specific  Heat. — Changes  of  Condition  at  different  Temperatures. — 
Latent  Heat. — Melting. — Vaporization. — The  Boiling  Point. — Spheroidal 
State.  —  Evaporation.  —  Condensation. — Solidification.  —  Freezing  by 
Liquefaction. — Freezing  by  Evaporation. — Regelation  .  .  328-335 

CHAPTER    IV. 

Dalton's  Laws. — Tension  of  different  Vapors. — Tension  in  Generator  and 
Condenser. — Tension  of  Steam. — Relations  of  Temperature,  Pressure, 
and  Volume  of  Steam. — The  Steam-engines  of  Savery  and  Newcomen. — 
Watt's  Engine. — Condensing  Engines. — Non-condensing  Engines. — 
Estimation  of  Steam  Power. — Mechanical  Equivalent  of  Heat  .  336-345 

CHAPTER    V. 

How  the  Air  is  warmed. — Limit  of  perpetual  Frost. — Isothermal  Lines. — 
Moisture  of  the  Atmosphere. — Temperature  and  Tension  of  Vapor. — 
Dew-point. — Hygrometers. — Dew,  Frost,  Fog,  and  Cloud. — Rain  and 
Snow. — Theories  of  Precipitation. — Cyclones.— Ventilation  of  Rooms. 
Sources  of  Heat  .  345-355 


PAET    VII.—  MAGNETISM. 

CHAPTER    I. 

Natural  and  Artificial  Magnet. — Attraction  of  Iron. — Polarity. — Action  of 
Magnets  on  each  other. — Distinction  between  Magnets  and  Magnetic 
Substances. — Magnetic  Induction. — Reflex  Influence. — Double  Induc- 
tions.—  Coercive  Force. — Magnetism  not  transferable. — Magnetic  In- 
tensity. —  Relation  of  Magnetic  Intensity  and  Distance.  —  Magnetic 
Curves. — Lines  of  Force,  Magnetic  Field. — Position  of  Equilibrium  of 
a  Needle  restrained  by  a  fixed  Axis.  —  Diamagnetism. — Influence  of 
the  surrounding  Medium. — Diamagnetic  Polarity. — Axis  in  Line  of 
greatest  Density.— Molecular  Changes 356-370 

CHAPTER    II 

Declination  of  the  Needle.— Isogonic  Curves.— Secular  and  Annual  Varia- 
tions.— Diurnal  Variation. — Dip  of  the  Needle. — Isoclinic  Curves. — 
Magnetic  Intensity  of  the  Earth.  —  Isodynamic  Curves.  —  Magnetic 
Charts. — Magnetic  Observatories. — Aurora  Borealis. — Source  of  the 
Earth's  Magnetism  .  .  370-378 


CONTENTS.  xj 

CHAPTER    III. 

PAGES 

Formation  of  Permanent  Magnets.— Compound  Magnets.— Preservation 
of  Magnets. — Saturation  of  Magnets. — Power  of  Magnets. — The  Decli- 
nation Compass.— The  Mariner's  Compass.— Astatic  Needle.— Theory 
of  Magnetism 378-384 

PART    VIIL— STATICAL    ELECTRICITY. 

CHAPTER    I. 

Elementary  Phenomena.— Common  Indications  of  Electricity.— The  Pen- 
dulum Electroscope. — Nature  of  Electricity. — Du  Fay's  Theory. — The 
Two  Electrical  States.— Mutual  Action  of  Electrified  Bodies.— Conduc- 
tion.—Modes  of  Insulating.— Various  Methods  of  developing  Electri- 
city .  .  . 385-390 

CHAPTER    II. 

The  Plate  Machine.— Cylinder  Machine.— Hydro-Electric  Machine.— 
Quadrant  Electrometer. — Phenomena  produced  by  Electric  Machines. — 
The  Torsion  Balance. — Law  of  Electrical  Force  and  Distance. — Electri- 
cal Unit.— Waste  of  Electricity  from  an  Insulated  Body.— The  Charge 
is  upon  the  Surface. — Potential.— Equipotential  Surfaces. — Capacity. — 
Distribution  of  Charge. — Surface  Density. — Rotation  by  Unbalanced 
Pressure.  ...  .  ,  .  .  .  .  .  .  390—399 

CHAPTER    III. 

Induction. — Successive  Actions  and  Reactions. — Induced  Charge  within 
a  Hollow  Conductor. — Division  of  the  Conductor. — Effect  of  lengthen- 
ing the  Conductor. — Disguised  Electricity.— Series  of  Conductors. — The 
Gold-leaf  Electroscope. — Mutual  Attractions  and  Repulsions  of  Bodies. 
— Franklin  Plate. — Condensing  Electroscope. — Influence  of  the  inter- 
posed Dielectric. — Leyden  Jar. — Spontaneous  Discharge. — Series  of 
Jars. — Division  of  a  Charge  in  any  given  Ratio. — Use  of  tb,e  Coatings. 
— Electrical  Vibrations. — Residuary  Charge. — The  Battery  of  Jars. — 
Discharging  Electrometers. — The  Effect  of  a  pointed  Conductor. — 
Electrophorus. — Dielectric  Machine. — Holtz  Machine  ,  .  400-415 

CHAPTER    IV. 

Effects  of  Electrical  Discharges.  —  Leichtenberg  Figures.  —  Luminous 
Figures. — Different  Routes  of  Discharge. — Chemical,  Magnetic,  and 
Physiological  Effects.— Velocity  of  Electricity  .  ,  .  .  416-431 

CHAPTER    V. 

Atmospheric  Electricity. — Thunder  Storms. — Identity  of  Lightning  and 
Electrical  Discharges. — Lightning  Rods. — Rules  for  the  Protection 
of  Buildings. — Protection  of  thie  Person. — How  Lightning  Causes 
Damage .  421-427 


xii  CONTENTS. 

PART    IX.— DYNAMICAL    ELECTRICITY. 

CHAPTEK    I. 

PAGB8 

Electricity  by  Chemical  Action.  —  An  Element. — A  Battery. — Volta's 
Pile. — Electricity  due  to  Contact. — Constant  Batteries. — Amalgamation 
of  Zinc. — Electromotive  Force. — Laws  of  Resistance. — Units  of  Resist- 
ance.— Strength  of  Current. — Ohm's  Law. — Modes  of  connecting  Bat- 
tery Elements. — Galvanic  and  Frictional  Electricity  compared  .  428-437 

CHAPTEE    II. 

Mutual  Action  of  Currents. — Continuous  Rotation. — Helices. — Solenoid. — 
Ampere's  Theory.— Relations  of  Currents  and  Magnets.— Galvanom- 
eters.— Polarity  of  the  Earth.  —  Thermo-electricity.  —  Tharmo-pile. — 
Magnetic  Induction  by  Currents. — The  Electro-magnet  -v  •""  .  438-450 

CHAPTER    III. 

Induced  Currents. — Characteristics. — Currents  induced  in  Coils. — Rulim- 
korfifs  Coil  .—Currents  induced  by  the  Motion  of  Conductors, — Lenz's 
Law.  — Magneto-electricity.  —  Arago's  Rotations.  —  Magneto-electric 
Machines. — The  Commutator. — Nollet's  and  Wild's  Machines. — Motion 
of  a  Straight  Conductor  through  a  Magnetic  Field. — Motion  of  a  Coiled 
Conductor. — Dynamo-electric  Machines  .  '  .  V  .  .'  450-464 

CHAPTER    IV. 

Electrolysis.— Voltameter.— Electro-plating.— Plante's  Cell.— Medicinal 
Vp plications  of  the  Current. — Electric  Lighting. — Heat  by  the  Current. 
— Electro-magnetic  Engines. — Electro-magnetic  Telegraph.— Resistance 
Coils. — Wheatstone's  Bridge.— Measurements  of  Resistance. — Duplex 
Telegraphy.  —  Atlantic  Telegraph  Cable. — Fire-alarm  Telegraph.  — 
Chronograph  . 465-481 


APPENDIX. 

APPLICATIONS    OF    THE    CALCULUS. 
I.  FALL  OF  BODIES. 

Differential  Equations.— Fall  through  small  Distances  near  the  Earth.— 
Through  great  Distances. — Method  of  finding  Velocity  and  Time. — Fall 
within  the  Earth.— Velocity  and  Time  found  .  .  ^  .  .  483-486 

II.  CEOTRE  OF  GRAVITY. 

Principle  of  Moments.— Formula?  prepared.— Applications  of  Formulae  to 
various  Cases  ...  .  486-490 


CONTENTS.  xiii 

III.  CENTRE  OF  OSCILLATION. 

PAGES 

Moment  of  Inertia  for  any  Axis. — Examples 490-493 

IV.  CENTRE  OF  HYDROSTATIC  PRESSURE. 
General  Formulae.— Examples 492-494 

V.  ANGULAR  RADIUS  OF  THE  RAINBOW  AND  THE  HALO. 

The  Primary  Bow.— The  Secondary  Bow.— The  Halo  .         ,        .        494-495 


(   UNIVERSITY  1 

X^LIFOHH^X 

NATURAL    PHILOSOPHY. 


INTRODUCTION. 

Art.  1.  Classification  of  Physical  Sciences.— The  ma- 
terial world  consists  of  two  parts — the  organized,  including  the 
animal  and  vegetable  kingdoms  ;  and  the  unorganized,  which 
comprehends  the  remainder.  Organized  matter  is  treated  of  in 
Physiology,  and  in  those  branches  of  science  usually  called  Natural 
History.  Unorganized  matter  forms  the  subject  of  Natural  Phi- 
losophy and  Chemistry.  Chemistry  considers  the  internal  consti- 
tution of  bodies,  and  the  relations  of  their  smallest  parts  to  each 
other.  Natural  Philosophy  deals  principally  with  the  external 
relations  of  bodies  and  their  action  upon  one  another.  If,  how- 
ever, the  bodies  are  so  large  as  to  constitute  ivorlds,  of  which  the 
earth  itself  is  one,  this  science  takes  the  name  of  Astronomy. 

The  word  Physics  is  much  used  to  include  both  Natural 
Philosophy  and  Chemistry ;  but  sometimes  it  is  applied  to  the 
branches  of  Natural  Philosophy,  except  Mechanics.  According 
to  the  latter  use  of  the  word,  Natural  Philosophy  is  divided  into 
two  general  subjects,  Mechanics  and  Physics. 

2.   Definitions  relating  to  Matter.— 

A  body  is  a  separate  portion  of  matter,  whether  large  or 
small. 

An  atom  is  a  portion  of  matter  so  small  as  to  be  indivisible. 

A  particle  denotes  the  smallest  portion  which  can  result 
from  division  by  mechanical  means,  and  consists  of  many  atoms 
united  together. 

A  Molecule  is  the  smallest  portion  of  any  substance  which  can 
exist  in  a  free  state,  and  is  made  up  of  atoms. 

Mass  is  the  quantity  of  matter  in  a  body,  and  is  usually 
measured  by  its  weight. 

Volume  signifies  the  space  occupied  by  a  body. 

Density  expresses  the  relative  mass  contained  within  a  given 


2  MECHANICS. 

volume.  Thus,  if  one  body  has  twice  as  great  a  mass  within  a  cer- 
tain volume  as  another  has,  it  is  said  to  have  twice  the  density. 

Pores  are  the  minute  portions  of  space  within  the  volume  of 
a  body,  which  are  not  filled  by  the  material  of  that  body.  All 
matter  is  porous,  some  kinds  in  a  greater  and  some  in  a  less  degree. 

Force  is  the  name  of  any  cause  which  gives,  or  tends  to  give, 
motion  to  matter,  or  which  changes,  or  tends  to  change,  motion 
already  existing. 

3.  Properties  of  Matter.— 

(1.)  Extension. — Every  portion  of  matter,  however  small,  has 
length,  breadth,  and  thickness,  and  thus  occupies  space.  This 
is  its  extension. 

(2.)  Impenetrability. — While  matter  occupies  space,  it  excludes 
all  other  matter  from  it,  so  that  no  two  atoms  can  be  in  exactly 
the  same  place  at  the  same  time.  This  property  is  called  im- 
penetrability. 

The  two  foregoing  are  often  called  essential  properties,  because 
we  cannot  conceive  matter  to  exist  without  them. 

(3.)  Divisibility. — Matter  is  divisible  beyond  any  known  limits. 
After  being  divided,  as  far  as  possible,  into  particles  by  mechani- 
cal methods,  it  may  be  still  further  reduced  by  chemical  action  to 
atoms,  which  are  too  small  to  be  in  any  way  recognized  by  the 
senses. 

(4.)  Compressibility. — Since  pores  exist  in  all  matter,  it  may 
be  compressed  into  a  smaller  volume.  Hence  all  matter  is  com- 
pressible, though  in  very  different  degrees. 

(5.)  Elasticity. — After  a  body  has  suffered  compression,  it 
shows,  in  some  degree  at  least,  a  tendency  to  restore  itself  to  its 
former  volume.  This  property  is  called  elasticity.  A  body  is  said 
to  be  perfectly  elastic  when  the  force  by  which  it  recovers  its  size 
is  equal  to  that  by  which  it  was  before  compressed.  The  word 
elasticity  is  used  generally  in  a  wider  sense  than  is  given  in  the 
above  definition,  namely,  the  tendency  which  a  body  has  to  recover 
its  original  form,  whatever  change  of  form  it  may  have  previously 
received.  Thus,  if  a  body  is  stretched,  bent,  twisted,  or  distorted 
in  any  other  way,  it  is  called  elastic,  if  it  tends  to  resume  its  form 
as  soon  as  the  force  which  altered  it  has  ceased.  Torsion  is  the 
name  of  the  elastic  force  which  tends  to  untwist  a  thread  or  wire 
when  it  has  been  twisted. 

(6.)  Attraction. — This  is  the  general  name  used  to  express  the 
universal  tendency  of  one  portion  of  matter  towards  another.  It 
receives  different  names,  according  to  the  circumstances  in  which 
it  acts.  The  attraction  which  binds  together  atoms  of  different 
kinds,  so  as  to  form  a  new  substance,  is  called  affinity,  and  is 


INTRODUCTION.  3 

discussed  in  Chemistry ;  that  which  unites  particles,  whether 
simple  or  compound,  so  as  to  form  a  body,  is  called  cohesion  ;  the 
clinging  of  two  kinds  of  matter  to  each  other,  without  forming  a 
new  substance,  is  called  adhesion  ;  and  the  tendency  manifested 
by  masses  of  matter  toward  each  other,  when  at  sensible  distances, 
is  called  gravity. 

(7.)  Inertia. — This  is  also  a  universal  property  of  matter,  and 
signifies  its  tendency  to  continue  in  its  present  condition  as  to 
motion  or  rest.  If  at  rest,  it  cannot  move  itself;  if  in  motion,  it 
cannot  stop  itself  or  change  its  motion,  either  in  respect  to  direc- 
tion or  velocity. 

4.  Branches  of  Natural  Philosophy. — Natural  Philosophy 
is  generally  divided  into  Mechanics,  Hydrostatics,  Pneumatics, 
Sound,  Magnetism,  Electricity,  Heat,  and  Light. 

Mechanics  treats  of  the  motion  and  equilibrium  of  bodies, 
caused  by  the  application  of  force.  Since  there  are  three  condi- 
tions of  matter,  solid,  liquid,  and  gaseous,  it  is  convenient  to 
divide  the  general  subject  of  Mechanics  into  three  branches. 

1st.  The  mechanics  of  solids,  also  called  Mechanics. 

3d.  The  mechanics  of  liquids,  called  Hydrostatics. 

3d.  The  mechanics  of  gases,  called  Pneumatics. 

All  the  other  branches  of  Natural  Philosophy  (often  called 
Physics)  treat  of  various  phenomena  caused  by  'minute  vibrations 
in  the  particles  of  matter.  These  vibrations  are  excited  in  differ- 
ent ways,  and  when  transmitted  to  us,  affect  one  or  more  of  our 
senses.  Thus,  sound  consists  of  such  vibrations  as  affect  the  sense 
of  hearing ;  and  light  is  another  mode  of  vibration,  that  affects 
only  the  sense  of  vision. 

It  was  formerly  customary  to  regard  magnetism,  electricity, 
heat,  and  light,  as  so  many  kinds  of  imponderable  matter,  that  is, 
matter  having  no  sensible  weight,  and  thus  distinguished  from 
solids,  liquids,  and  gases,  which  are  the  different  forms  of  ponder- 
able matter.  But  it  is  now  known  that  when  forces  are  applied  to 
matter,  they  not  only  produce  the  visible  forms  of  motion,  but 
may  be  made  to  develop  either  sound,  magnetism,  electricity,  heat, 
or  light ;  and  that  most  of  these  modes  of  motion  may  be  trans- 
formed into  others,  and  each  may  be  made  a  measure  of  the  force 
which  is  employed  to  produce  it. 


OF  THE 

UNIVERSITY 


PART   I. 


CHAPTER  I. 

MOTION    AND    FORCE. 

5.  Classification  of  Motions. — Motion  is  change  of  place, 
and  is  either  uniform  or  variable.    In  uniform  motion  equal 
spaces  are  passed  over  in  equal  times,  however  small  the  times 
may  be.     In  variable  motion  the  spaces  described  in  equal  times 
are  unequal.     Such  motion  may  be  either  accelerated  or  retarded. 
In  accelerated  motion  the  spaces  described  in  equal  times  become 
continually  greater  ;  in  retarded  motion  they  become  continually 
less.    Motion  is  said  to  be  uniformly  accelerated  if  the  increments 
of  space  in  equal  times  (however  small)  are  equal;  and  uniformly 
retarded  if  the  decrements  are  equal. 

Velocity  is  the  space  described  in  the  unit  of  time.  In  Me- 
chanics, one  second  is  much  used  as  the  unit  of  time,  and  one  foot 
as  the  unit  of  space ;  hence,  velocity  is  the  number  of  feet  de- 
scribed in  one  second. 

6.  Uniform  Motion. — When  motion  is  uniform,  the  number 
of  feet  described  in  one  second,  multiplied  by  the  number  of 
seconds,  obviously  gives  the  whole  space.    Let  s  =  space,  t  =  time, 

and  v  =  velocity  ;  then  s  =  t  v  ;  .-.  t  =  -,  and  v  =  -•     If  this 

v  t 

space  is  compared  with  another,  s',  described  in  the  time  t',  with 
the  velocity  v',  then  s  :  s' : :  t  v  :  t'  v' ;  or  briefly,  in  the  form  of  a 

o  o 

variation,  s  oc  t  v.     In  like  manner  t  oc  -,  and  v  oc  -• 

v  t 

If  two  bodies,  moving  uniformly,  describe  equal  spaces,  then 
s  =  s';  .-.t  v  —  t'  v' ;  .-.  t  :  t'  ::  v'  :  v.  That  is,  in  order  that 
two  bodies  may  describe  equal  spaces,  their  velocities  must  vary 
inversely  as  the  times  during  which  they  move. 

7-  Questions  on  Uniform  Motion. — 

1.  A  ball  was  rolled  on  the  ice  with  a  velocity  of  78  feet  per 
second,  and  moved  uniformly  21  seconds ;  what  space  did  it  de- 
scribe? Ans.  1638  feet. 


MOMENTUM.  5 

2.  A  steamboat  moved  uniformly  across  a  lake  17  miles  wide, 
at  the  rate  of  20  feet  per  second ;   what  time  was  occupied  in 
crossing  ?  Ans.  Ih.  14m.  48s. 

3.  On  the  supposition  that  the  earth  describes  an  orbit  of  600 
millions  of  miles  in  365J  days,  with  what  velocity  does  it  move 
per  second  ?  Ans.  19  miles,  nearly. 

4.  Three  planets  describe  orbits  which  are  to  each  other  as  15, 
19,  and  12,  in  times  which  are  as  7,  3,  and  5  ;  what  are  their  rela- 
tive velocities  ?  Ans.  225,  665,  and  252. 

8.  Momentum. — The  product  of  the  mass  of  a  body  and  its 
velocity  is  called  momentum.     Thus  let  m  =  momentum,  q  =  the 

mass,  and  v  =  the  velocity,  and  we  have  m  =  q  v,  q  =  — ,  and 

m 

v  =  — 

q 

If  the  momentum  of  one  body  equals  that  of  another,  then, 
since  m  —  m',  q  v  =  q'  v',  .'.  q  :  q' ::  v' :  v.  That  is,  in  order  that 
the  momenta  of  two  bodies  should  be  equal,  their  masses  must 
vary  inversely  as  their  velocities. 

Since  there  are  two  elements  entering  into  the  momentum  of  a 
body — namely,  .its  mass,  usually  expressed  in  pounds,  and  its 
velocity,  expressed  in  feet  per  second — therefore  momentum  can- 
not be  measured  either  in  pounds  or  in  feet,  being  in  nature  unlike 
either.  The  word  foot-pound  is  employed  for  the  unit  of  momen- 
tum whenever  the  unit  of  mass  is  a  pound  and  the  unit  of  velocity 
is  a  foot  per  second. 

9.  Questions  on  Momentum.— 

1.  A  ship  weighing  336,000  Ibs.  is  dashed  against  the  rocks  in 
a  storm,  with  a  velocity  of  16  miles  per  hour  ;  with  what  momen- 
tum did  she  strike  ?  Ans.  7,884,800  foot-pounds. 

2.  A  ball  weighing  1  oz.  is  fired  into  a  log  weighing  53  Ibs., 
suspended  so  as  to  move  freely,  and  imparts  a  velocity  of  2  ft.  per 
second.     Assuming  that  the  log  and  ball  have  a  momentum  equal 
to  the  previous  momentum  of  the  ball  alone,  required  the  velocity 
of  the  ball.  Ans.  1,698  ft.  per  second. 

3.  Suppose  a  comet,  whose  velocity  is  1,000,000  miles  per  hour, 
has  the  same  momentum  as  the  earth,  whose  velocity  is  19  miles 
per  second  ;  what  is  the  ratio  of  their  masses  ?       Ans.  1  :  14.6. 

4.  Two  railway  cars  have  their  quantities  of  matter  as  7  to  3, 
and  their  momenta  as  8  to  5  ;  what  are  their  relative  velocities  ? 

Ans.  As  24  to  35,  or  nearly  5  to  7. 

5.  The  momentum  of  a  cannon-ball  was  434  foot-pounds ; 
what  must  be  the  velocity  of  a  half-ounce  bullet,  in  order  to  have 
the  same  momentum  ?  Ans.  13,888  feet. 


6  MECHANICS. 

10.  Classification    of   Forces. — The  principal  forces  in 
nature  are  the  following  : 

1.  Attraction  in  its  several  forms.     Cohesion  and   chemical 
affinity  are  the  forces  which  bind  together  the  particles  and  atoms 
of  bodies,  and  gravity  is  that  which  everywhere  near  the  earth 
causes  bodies  to  fall  toward  it,  or  to  press  upon  it. 

2.  Elasticity. — This  is  a  force  which,  in  many  kinds  and 
conditions  of  matter,   tends   to   repel   the   particles   from   each 
other. 

The  forces,  whether  attraction  or  repulsion,  which  exist  among 
the  atoms  or  molecules  of  a  body,  are  called  molecular  forces. 

3.  Muscular  force. — All  living  beings  are  endowed  with  this 
force,  by  which  they  put  in  motion  bodies  around  them,  and  by 
acting  upon  other  bodies,  are  enabled  also  to  move  themselves 
from  place  to  place. 

4.  Matter  in  motion. — If  a  body  which  some  force  has  put  in 
motion  impinges  on  another  body,  it  imparts  motion  to  it,  and  is 
therefore  itself  a  force.     This  is  true  not  only  of  ordinary  visible 
motions,  but  of  those  small  and  often  invisible  vibrations,  which 
manifest  themselves  as  sound,  heat,  &c.     Gravity,  or  any  other 
force,  may  cause  heat,  and  heat  may  cause  light  and  electricity. 
Thus,  any  form  of  motion  is  a  force,  and  it  can  be  employed  to 
produce  other  forms. 

11.  Impulsive  and  Continued   Forces  and  their  Ef- 
fects.— An  impulsive  force  is  one  which  has  no  sensible  continu- 
ance, as  the  blow  of  a  hammer.     A  continued  force  is  one  which 
acts  during  a  perceptible  length  of  time.     Continued  forces  are 
subdivided  into  constant  and  variable.     A  constant  force  has  the 
same  intensity  during  the  whole  time  of  its  action ;  a  variable 
force  is  one  whose  intensity  changes. 

Keeping  in  mind  the  property  of  inertia,  we  associate  different 
kinds  of  motion  with  the  forces  which  produce  them,  as  follows  : 

1.  An  impulsive  force  causes  uniform  motion. 

2.  A  continued  force,  accelerated  motion. 

3.  A  constant  force,  uniformly  accelerated  motion. 

4.  A  variable  force,  unequally  accelerated  motion. 

If  the  force  is  applied  in  a  direction  opposite  to  that  in  which 
the  body  has  a  previous  uniform  motion,  the  connection  is  the 
following  : 

5.  An  impulsive  force  causes  uniform  motion,  or  rest. 

6.  A  continued  force,  retarded  motion. 

7.  A  constant  force,  uniformly  retarded  motion. 

8.  A  variable  force,  unequally  retarded  motion. 

In  cases  1  and  5,  it  is  obvious  that,  the  impulse  being  given, 


MOTION    AND    FORCE.  7 

the  body  is  left  to  itself,  and  cannot  change  the  state  of  motion  or 
rest  impressed  on  it. 

In  2,  3,  and  4,  it  must  be  considered  that  the  force  at  each 
instant  adds  a  new  increment  to  the  uniform  motion  which  the 
body  would  have  had  if  the  force  had  ceased  ;  and  if  the  force  is 
constant,  those  increments  are  equal ;  if  variable,  they  are  un- 
equal. 

In  6,  7,  and  8,  the  same  statements  may  be  made  in  regard  to 
decrements.  It  is  also  plain  that  in  these  three  last  cases,  if  the 
force  continues  to  act  indefinitely,  the  motion  will  be  retarded 
until  the  body  comes  to  a  state  of  momentary  rest,  and  then  is 
accelerated  in  the  direction  of  the  force. 

12.  Measure  of  Force. — The  intensity  of  an  impulsive  force 
is  measured  by  the  momentum  which  it  will  produce  or  destroy  ; 
that  is,  /  oc  m.  But  m  <x  q  v  ;  /.  /  oc  q  v.  Hence,  if  q  is  con- 
stant, /  oc  v.  If,  then,  an  impulse  is  applied  to  a  given  mass,  the 
intensity  of  that  impulse  is  measured  by  the  velocity  which  it 
imparts  or  destroys. 

Thus,  if  a  force  gives  to  a  mass  q  a  velocity  v,  the  same  force 

would  give  to  one-half  the  mass,  or  |  a  velocity  2  v  ;  for  /  =  q  ?% 

a 

and  also  /  =  f  x  2  v.    So  also  if  a  force  /  gives  a  velocity  v  to  a 
/c 

mass  q,  a  force  2/  would  be  required  to  give  to  the  same  mass  a 
velocity  2  v. 

But  in  the  case  of  a  constant  force,  the  momentum  depends 
not  only  on  the  intensity  of  the  force,  but  on  the  time  during 

in 

which  it  is  applied  ;  that  is,  ft  ^~m,  and/  oc  —  •    If  the  mass  of 

t 

the  body  is  given,  then,  as  in  the  case  of  an  impulsive  force, 

q  being  constant,  ftccv,  and  /  oc  -• 

t 

A  constant  force  produces  uniformly  accelerated  motion, 
since  the  increase  of  velocity  during  any  unit  of  time  is  constant. 
The  unit  of  time  usually  chosen  is  one  second  and  the  increase  in 
velocity  per  second  is  called  the  acceleration  due  to  the  force.  If 
a  force /produces  an  acceleration  of  20  ft.  per  second,  and  another 
force  /'  produces  an  acceleration  of  40  ft.  per  second,  the  mass 
being  supposed  the  same  in  both  cases,  the  force/'  ="2/;  or  if 
a  force  /  acting  upon  a  body  at  rest,  gradually  increases  its  ve- 
locity from  0  to  20  ft.  at  the  end  of  the  second,  and  if  /'  will 
increase  the  velocity  of  the  same  body  from  0  to  40  ft.  at  the  end 
of  the  second,  then/'  =  2fu 

If  a  force  /  gives  to  a  mass  q  an  acceleration  of  50  ft,  and 


8  MECHANICS. 

force  /'  gives  to  mass  3  q  an  acceleration  50  ft.,  then  /'  =  3/1 
If  a  force  f  gives  to  a  mass  q  an  acceleration  30  ft.,  and  /'  gives 
to  2  q  an  acceleration  60  ft.,  then/'  =  4/. 

To  express  the  measure  of  a  variable  force,  let  t  be  a  constant 
and  infinitely  small  portion  of  time  ;  then  the  force  varies  as  the 
mass  multiplied  by  the  increment  of  velocity  imparted  in  that 
time  divided  by  the  time. 

13.  The  Three  Laws  of  Motion. — All  the  phenomena  of 
motion  in  Mechanics  and  Astronomy  are  found  to  be  in  accord- 
ance with  three  first  principles,  which  Newton  announced  in  his 
Principia,  and  which  are  to  be  regarded  as  forming  the  basis  of 
mechanical  science.  They  may  be  named  and  defined  as  follows  : 

1.  The  law  of  inertia. — A  body  at  rest  tends  to  remain  at  rest ; 
and  a  body  in  motion  tends  to  move  forever,  in  a  straight  line, 
and  uniformly. 

2.  The  law  of  the  coexistence  of  motions. — If  several  motions 
are  communicated  to  a  body,  it  will  ultimately  be  in  the  same 
position,  whether  those  motions  are  simultaneous  or  successive. 

3.  The  law  of  action  and  reaction. — If  any  kind  of  action  takes 
place  between  two  bodies,  it  produces  equal  momenta  in  opposite 
directions  ;  or,  every  action  is  accompanied  by  an  equal  and  oppo- 
site reaction. 

The  truth  of  these  laws  cannot  be  established,  except  approxi- 
mately, by  direct  experiments,  because  gravity,  friction,  and  the 
resistance  of  air,  interfere  more  or  less  with  every  possible  experi- 
ment. They  are  to  be  learned  rather  by  a  careful  study  of  the 
phenomena  of  motion  in  general.  We  see  an  approximation  to 
the  first  law,  in  rolling  a  ball  on  a  horizontal  surface ;  first,  on 
the  earth,  then  on  a  floor,  and  again  on  smooth  ice,  the  motion 
approaching  toward  uniformity  as  obstructions  are  diminished,  and 
gravity  producing  no  direct  effect,  because  acting  at  right  angles 
to  the  line  of  motion.  The  discussion  of  the  second  law  is  reserved 
for  Chapter  III.  The  third  law  is  illustrated  by  a  variety  of  cases 
in  collision,  attraction,  and  repulsion.  Suppose  that  a  body  A, 
being  in  motion,  strikes  directly  against  J9,  which  is  at  rest ;  it  is 
found  that  B  acquires  a  certain  momentum,  and  that  A  loses  (that 
is,  acquires  in  an  opposite  direction)  an  equal  amount.  The  same 
is  true  if  B  is  in  motion,  and  A  either  overtakes  or  meets  it.  In 
the  collision  of  two  railroad  trains,  it  is  immaterial  as  to  the 
effects  which  they  will  respectively  suffer,  whether  each  is  moving 
towards  the  other,  or  whether  one  is  at  rest,  provided  that  in  the 
latter  case  the  moving  train  has  a  momentum  equal  to  the  mo- 
menta of  the  two  trains  in  the  former  case.  When  a  magnet 
attracts  a  piece  of  iron,  each  moves  towards  the  other  with  the 


FORCE    OF    GRAVITY.  9 

same  momentum.  A  spring  between  two  bodies  A  and  B  drives 
A  from  B  with  as  much  momentum  as  B  from  A  ;  and  the  sud- 
den expansion  of  burning  gunpowder,  which  propels  the  balls 
when  a  broadside  is  fired,  causes  an  equal  amount  of  motion  of 
the  ship  in  the  opposite  direction. 

14.  Force   of  Gravity. — Every  mass  of  matter  near  the 
earth,  when  free  to  move,  pursues  a  straight  line  towards  its 
centre.     The  force  by  which  this  motion  is  produced  is  called 
gravity  ;  either  the  gravity  of  the  body  or  the  gravity  of  the  earth ; 
for  the  attraction  is  mutual  and  equal,  in  accordance  with  the 
third  law  of  motion.    It  is  easy  to  understand  why  a  small  mass 
should  attract  a  large  one,  as  much  as  the  large  mass  attracts  the 
small  one.     Let  A  consist  of  one  atom  of  matter,  and  B,  at  any 
distance  from  it,  consist  of  ten  atoms.    If  it  be  admitted  that  A 
attracts  one  atom  of  B  as  much  as  that  one  atom  attracts  A,  then 
the  above  conclusion  follows.     For  A  attracts  each  of  the  ten 
atoms  of  B  as  much  as  each  of  the  same  ten  attracts  A  ;  so  that  A 
exerts  ten  units  of  attraction  on  B,  while  B  exerts  ten  units  of 
attraction  on  A.     The  same  reasoning  obviously  applies  to  the 
earth  in  relation  to  the  small  bodies  on  its  surface. 

15.  Relation  of  Gravity  and  Mass.— At  the  same  dis- 
tance from  the  centre  of  the  earth,  gravity  varies  as  the  mass. 
This  is  because  it  operates  equally  on  every  atom  of  a  body ; 
hence  the  greater  the  number  of  atoms  in  a  body,  the  greater  in 
the  same  ratio  is  the  attraction  exerted  upon  it.     That  gravity 
varies  as  the  mass  is  also  proved  from  the  observed  fact,  that  in  a 
vacuum  it  gives  the  same  velocity,  in  the  same  time,  to  every 
mass,  however  great  or  small,  and  of  whatever  species  of  matter. 
For  a  constant  force,  acting  for  a  given  time,  is  measured  by  the 
momentum  which  it  produces  (Art.  12),  and  that  momentum,  if 
the  velocity  is  the  same,  varies  as  the  mass ;  therefore  the  force 
also  varies  as  the  mass  to  which  it  imparts  the  given  velocity. 

If  a  body  is  not  free  to  move,  its  tendency  towards  the  earth 
causes  pressure  ;  and  the  measure  of  this  pressure  is  called  the 
iveight  of  the  body.  Weight  is  usually  employed  as  a  measure  of 
the  mass  in  bodies.  The  foregoing  relations  are  embodied  in  the 
following  expressions  :  g  oc  q  ;  and  w  oc  q. 

16.  Relation  of  Gravity  and  Distance.— At  different  dis- 
tances above  the  earth's  surface,  gravity  varies  inversely  as  the 
square  of  the  distance  from  the  centre. 

The  demonstration  of  this  proposition  is  reserved  for  astron- 
omy, where  it  is  shown  by  the  movements  of  the  bodies  in  the 
solar  system  that  this  law  applies  to  them  all. 


10  MECHANICS. 

The  moon  is  60  times  as  far  from  the  earth's  centre  as  the  dis- 
tance from  that  centre  to  the  surface  :  therefore  the  attraction  of 
the  earth  upon  the  particles  of  the  moon  is  3600  times  less  than 
upon  particles  at  the  surface  of  the  earth.  At  the  height  of  4000 
miles  above  the  earth,  gravity  is  four  times  less  than  at  the  sur- 
face. But  the  heights  at  which  experiments  are  commonly  made 
upon  the  weights  of  bodies  bear  so  small  a  ratio  to  the  radius  of 
the  earth,  that  this  variation  is  commonly  imperceptible.  At  the 
height  of  half  a  mile,  the  diminution  does  not  amount  to  more 
than  about  ^wth  Part  °f  ^ne  weight  at  the  surface.  For,  let 
r  =  the  radius  of  the  earth  =  4000  miles,  nearly  ;  and  let  x  be 
the  height  of  the  body,  w  its  weight  at  the  earth's  surface,  and 
w'  its  weight  at  the  height  x.  Then, 

w:w'::    r  +  x    :  r2  ::  r2  +  2rx  +  &  :  r2. 


w:  w—w'  ::  rt  +  Zrx+x*  :  2rx  +  x*  .:  w—w'  — 


But  when  #  is  a  small  fraction  of  r,  x*  may  be  neglected,  and 

.    ,  ,       wx2x 

the  formula  becomes  w  —  w  =  -  ......     (B). 

r-\-%x 

Let  x  be  half  a  mile;  then  =  j-gVrth  part  of  the  whole 

weight  ;  or,  a  body  would  weigh  so  much  less  at  the  height  of  half 
a  mile  than  at  the  surface  of  the  earth.  But  if  the  height  were  as 
great  as  100  miles  above  the  earth,  the  loss  should  be  calculated 
by  formula  (A),  since  the  other  would  give  a  result  too  small  by 
one  per  cent,  or  more,  according  to  the  height. 

What  loss  of  weight  would  a  body  sustain  by  being  elevated  500 
miles  above  the  earth?  Am.  ^J,  or  more  than  -J-  of  its  weight. 

The  relation  of  gravity  to  distance  is  expressed  by  the  formula 

g  oc  -j£  ;   and  as  g  <x  q  also,  it  varies  as  the  product  of  the  two  ; 

that  is,  g  oc  ~  ;  or  gravity  towards  the  earth  varies  as  the  mass  of 

the  body  directly,  and  as  the  square  of  the  distance  from  the  earth9  s 
centre  inversely. 

17.  Gravity  within  a  Hollow  Sphere.—  A  particle  situated 
within  a  spherical  shell  of  uniform  density,  is  equally  attracted  in 
all  directions,  and  remains  at  rest.  This  is  true,  because,  in  every 
direction  from  the  particle,  the  mass  varies  at  the  same  rate  as  the 
square  of  the  distance,  so  that  attraction  increases  for  one  reason, 
as  much  as  it  diminishes  for  the  other  ;  which  is  proved  as  follows  : 

Let  the  particle  P  (Fig.  1)  be  at  any  point  within  the  spherical 
shell  A  B  CD.  Let  two  opposite  cones  of  revolution,  of  very 
small  angle,  have  their  vertices  at  P,  and  suppose  the  figure  to  be 


FORCE    OF    GRAVITY. 


11 


FIG.  1. 


a  section  through  the  centre  of  the  sphere  and  the  axis  of  the 
cones.  Then  A  B  and  a  b  will  be  the  major  axes  of  the  small 
ellipses,  which  are  the  bases  of  the  cones,  and 
which  may  be  considered  as  plane  figures.  By 
geometry,  A  P  :  P  B  : :  P  b  :  P  a  ;  and  the 
angles  at  P  being  equal,  the  triangles  are  simi- 
lar ;  hence  the  angles  B  and  a  are  equal. 
Therefore,  the  bases  of  the  cones  are  similar 
ellipses,  being  sections  of  similar  cones,  equally 
inclined  to  the  sides.  By  similar  triangles, 

~B* :  ~ab2. 


P  b* : :  A  B* :  a  ff.     Let  q  and  q'  repre- 
sent the  masses  of  the  thin  laminae  which  form  the  bases  ;  then, 
since  similar  ellipses  are  to  each  other  as  the  squares  of  their  major 


axes,  we  have 


q:q' 


or  rrrf^  = 


q' 


But 


represent  the  attractions  of  the  bases  respec- 


FIG.  2. 


AP*  Ptf 

tively  on  the  particle  (Art.  16)  ;  and  since  these  are  equal,  the 
particle  is  equally  attracted  by  all  the  opposite  parts  of  the  spheri- 
cal shell. 

18.  Gravity  within  a  Solid  Sphere.— Within  a  solid  sphere 
of  uniform  density,  weight  varies  directly  as  the  distance  from  the 
centre. 

Let  a  particle  P  (Fig.  2)  be  within  the  solid 
sphere  A  D  C ;  and  call  its  distance  from  the 
centre  d.  Now,  by  the  preceding  article  the 
shell  exterior  to  it,  ADR,  exerts  no  influence 
upon  it,  and  it  is  attracted  only  by  the  sphere 
P  R  8.  Let  q  represent  the  quantity  of  this 

sphere  ;  then  gravity  varies  as  —••    But  q  a  d8 ; 

a 
^73 
.*.  g  oc  -^  oc  d.    Hence,  in  the  earth  (if  it  be  supposed  spherical 

and  uniformly  dense,  though  it  is  neither  exactly),  a  body  at  the 
depth  of  1000  miles  weighs  three-fourths  as  much  as  at  the  sur- 
face, and  at  2000  miles  it  weighs  half  as  much,  while  at  the 
centre  it  weighs  nothing. 

Comparing  this  proposition  with  Art.  16,  we  learn  that  just  at 
the  surface  of  the  earth  a  body  weighs  more  than  at  any  other 
place  without  or  within.  Within,  the  weight  diminishes  nearly 
as  the  distance  from  the  centre  diminishes  ;  without,  it  diminishes 
as  the  square  of  the  distance  from  the  centre  increases. 

At  the  surface  of  spheres  having  the  same  density,  weight  varies 
as  the  radius  of  the  sphere.  Let  r  be  the  radius  of  the  sphere,  and 


12  MECHANICS. 

Q  r3 

q  its  mass ;  then,  since  g  oc  -^-,  in  this  case  it  varies  as  -j  oc  r. 

Therefore,  if  two  planets  have  equal  densities,  the  weight  of  bodies 
upon  them  is  as  their  radii  or  their  diameters.  If  a  ball  two  feet 
in  diameter  has  the  same  density  as  the  earth,  a  particle  of  dust 
at  its  surface  is  attracted  by  it  nearly  21  millions  of  times  less 
than  it  is  by  the  earth. 

19.  Questions  for  Practice.— 

1.  How  much  weight  would  a  rock  that  weighs  ten  tons 
(22,400  Ibs.)  at  the  level  of  the  sea,  lose  if  elevated  to  the  top  of 
a  mountain  five  miles  high  ?  Ans.  55.8952  Ibs. 

2.  If  the  earth  were  a  hollow  sphere,  and  if,  through  a  hole 
bored  through  the  centre,  a  man  were  let  down  by  a  rope,  would 
the  force  required  to  support  him  be  increased  or  diminished  as 
he  descended  through  the  solid  crust,  and  where  would  it  become 
equal  to  nothing? 

3.  How  much  would  a  44-pound  shot  weigh  at  the  centre  of 
the  earth ;  how  much  at  a  point  half  way  from  the  centre  to  the 
surface ;  and  how  much  100  miles  below  the  surface  ? 

4.  Suppose  a  32-pound  cannon-ball,  fired  with  the  velocity  of 
2000  feet  per  second,  to  have  the  same  momentum  as  a  battering- 
ram  whose  weight  is  5760  pounds;  find  the  velocity  of  the  latter. 

Ans.  11.11  ft.  per  sec. 

5.  Suppose  light  to  have  weight,  and  one  grain  of  it  moving  at 
the  rate  of  192,000  miles  per  second,  to  impinge  directly  against  a 
mass  of  ice  moving  at  the  rate  of  1.45  feet  per  second,  and  to  stop 
it ;  required  the  weight  of  the  ice. 

Ans.  99877.832  Ibs.,  or  nearly  44J  tons,  reckoning  7000  gr.=l  Ib. 

6.  If  a  ball  of  the  same  density  with  the  earth,  -fath  of  a  mile 
in  diameter,  were  to  fall  through  its  own  diameter  toward  the 
earth,  what  space  would  the  earth  move  through  to  meet  the  ball, 
the  diameter  of  the  earth  being  taken  at  8000  miles  ? 

Ans.   soooobinnnr  incn»  nearly. 

7.  If  a  hole  were  bored  through  the  centre  of  the  earth,  what 
would  be  the  conditions  of  the  motion  of  a  stone  dropped  into 
the  hole  ? 

In  its  descent  towards  the  centre,  the  force  of  gravity  would 
continually  decrease  till  at  the  centre  it  became  zero  ;  but  though 
this  force  decreases  in  intensity,  it  will  at  each  instant  increase  the 
previously  existing  velocity,  though  by  decreasing  increments,  so 
that  the  stone  will  have  its  greatest  velocity  at  the  centre  of  the 
earth  :  it  will  then,  in  an  inverse  order,  suffer  continually  increas- 
ing decrements  of  velocity  until  it  finally  comes  to  rest  at  the  other 
surface  of  the  earth,  when  it  will  return  under  similar  condition". 


CHAPTER  II. 

VARIABLE  MOTION.  — CONSTANT  FORCES. 

20.  Relation  of  Time  and  Acquired  Velocity.— When  a 

body  moves  with  uniform  motion,  S  —  t  v  (Art.  6).  When  a 
body  moves  with  uniformly  varied  motion  the  case  is  somewhat 
different. 

Let  us  consider  the  case  of  a  body  that  moves  with  uniformly 
increasing  velocity.  Suppose  the  body  to  start  from  rest  and  at 
the  end  of  the  1st  second  to  have  acquired  a  velocity  of  10  ft.  per 
second;  that  is  to  say,  a  velocity  which  would  carry  it  over  10  ft. 
per  second  during  the  next  and  each  succeeding  second,  if  the 
force  ceased  to  act  at  the  end  of  the  first  second.  Now  since  the 
velocity  is  supposed  to  increase  uniformly,  we  shall  have  at  the 
end  of  the  2d  second  a  velocity  of  20  ft.,  at  the  end  of  the  3d  a 
velocity  of  30  ft.,  and  so  on. 

Hence  the  first  law  of  motion  under  the  action  of  a  constant 
force :  In  uniformly  accelerated  motion  the  acquired  velocities 
vary  as  the  times. 

21.  Space  Passed  Over. — Since  the  body  started  from  rest 
and  gained  uniformly  in  velocity  till  it  acquired  a  velocity  of 
50  ft.  per  second  at  the  end  of  the  5th  second,  it  is  evident  that  its 
average  velocity  was  25  ft.  per  second  ;  for  at  the  start  its  velocity 
was  0  and  at  the  end  was  50  ;  at  an  interval  of  one  second  after 
starting  it  had  a  velocity  of  10,  and  one  second  before  the  end  of 
the  time  considered  it  had  a  velocity  of  40  ;  two  seconds  after  start- 
ing the  velocity  was  20,  and  two  seconds  before  the  end  of  the 
time  the  velocity  was  30.     Thus  the  less  velocity  at  any  given 
interval  is  balanced  by  the  greater  velocity  during  the  correspond- 
ing interval  of  the  pair,  and  we  are  thus  enabled  to  find  the  dis- 
tance passed  over,  by  multiplying  the  average  velocity,  of  25  ft. 
per  second,  by  the  time,  5  seconds,  giving  the  space  125  feet. 

22.  Space  Described  during  ist  Second.-^-We  have  con- 
sidered the  velocities  at  intervals  of  one  second,  but  we  could 
have  chosen  smaller  intervals  as  well,  and  no  matter  how  small 
we  make  our  unit  of  time,  the  law  holds  good.     Now,  during  the 
first  second  the  body  acquired  a  velocity  of  10  ft,  and  if  we  sup- 
pose the  first  second  to  be  divided  into  10  equal  intervals,  we  may 


14 


MECHANICS. 


apply  the  same  analysis  to  these  as  to  the  five  full  seconds  already 
considered;  and  we  find  the  average  velocity  to  be  — ^— ,  or 

A 

5  ft. :  hence,  since  the  body  moved  for  one  second  with  a  velocity 
which  would  average  5  ft.  per  second,  it  must  have  moved  over 
5  ft. ;  hence,  a  body  starting  from  rest  ivill,  under  the  action,  of  a 
constant  force,  move  during  the  first  second  over  a  space  equal  to 
one-half  the  velocity  acquired  at  the  end  of  that  second. 

23.  Space   Described  during  any  Second. — The  space 
described  during  any  second  is  one-half  the  velocity  impressed 
upon  the  body  by  the  constant  force  during  that  second,  plus  the 
space  described  by  reason  of  velocity  already  impressed  upon  the 
body  by  previous  action  of  the  force. 

24.  Relations  of  Time,  Space,  and  Acquired  Velocity. 

— It  is  necessary  to  know  all  the  possible  relations  between  the 
space,  time,  and  acquired  velocities.  Let  us  now  examine  the 
relations  between  time  and  space.  During  the  first  second  the 
body,  in  the  case  already  given,  moves  over  5  ft.  and  acquires  a 
velocity  of  10  ft. ;  during  the  2d  second  it  will  move  over  10  ft. 
in  consequence  of  the  velocity  already  impressed,  and  over  5  ft. 
additional  because  of  the  continued  action  of  the  force,  making  a 
total  of  15  ft.  At  the  beginning  of  the  3d  second  the  velocity  is 
20  ft.,  and  the  body  will  move  over  20  ft.  in  consequence  of  this, 
together  with  5  ft.  more  on  account  of  the  continued  action  of 
the  force ;  and  so  on  to  the  end  of  the  time. 
Hence,  we  have — 


Times. 

Ac.  vel.  at  begin- 
ning of  interval. 

Spaces  described 
during  interval. 

Total  spaces. 

1st  Sec. 

0 

5 

For  1  sec.      5 

2d      " 

10 

15 

"    2    "      20 

3d      " 

20 

25 

"    3    "      45 

4th    " 

30 

35 

"    4    "      80 

5th    " 

40 

45 

"    5    "     125 

Examining  the  above  results,  and  calling  the  space  described 
during  the  1st  second  S,  we  have  the  space  during 

2  sees.  =  S  x    4  =  #  x  22 

3"  =Sx9  =  Sx& 

4  "  =  S  x  16  =  8  x  42 

5  "  =  S  x  25  =  S  X  52 

That  is  to  say,  The  spaces  described  under  the  action  of  a  constant 


VARIABLE    MOTION.— CONSTANT    FORCES.         15 

force  are  proportional  to  the  squares  of  the  times  during  which 
the  force  acts. 

Acquired  velocities  are  proportional  to  the  times,  and  therefore 
the  spaces  must  be  also  proportional  to  the  squares  of  the  acquired 
velocities. 

25.  Laws  of  Uniformly  Accelerated  Motion. — To  re- 
capitulate ;  when  bodies  move  under  the  action  of  a  constant 
force,  the  following  relations  exist  between  space,    time,  and 
velocity : 

1.  The  acquired  velocities  vary  as  the  times. 

2.  The  spaces  vary  as  the  squares  of  the  times. 

3.  The  spaces  vary  as  the  squares  of  the  acquired  velocities. 

As  an  aid  to  the  memory,  the  following  analogy  may  be  em- 
ployed. Let  s  be  the  space  described,  v  the  velocity  acquired  by 
a  body  moving  from  rest  for  the  time  t,  s'  the  space  described, 
v'  the  velocity  acquired  at  any  other  period  t' ;  then,  from  what  has 
already  been  demonstrated,  if  t  and  t'  be  represented  by  the  lines  A  B 
and  A  D  (Fig.  3),  and  v  and  v'  by  the  lines  B  C  and 

D  E,  drawn  at  right  angles  to  them,  s  and  s'  will 
be  represented  by  the  triangles  ABC  and  A  D  E. 
For  A  B  C  :  A  D  E  : :  A  B*  :  A  D* ;  or,  as 
B  C2  :  D  E*\  or  s  :  s'  ::  P  :  t\  or  as  ^  :  v\ 
The  velocity  acquired  varies  as  the  time ;  from 
the  similar  triangles  A  B  0,  A  D  E,  we  have 
B  C  :  D  E  : :  A  B  :  A  D,  or,  v  :  v'  :  :  t  :  t'. 

26.  Formulae. — Let  us  represent  by  f  the  acceleration  due  to 
a  force,  that  is  to  say,  the  increase  of  velocity  per  second  due  to 
the  action  of  the  force ;  then  the  space  passed  over  during  the 
1st  second,  if  starting  from  rest,  would  be  £/,  as  deduced  in 
Art.  22.     Calling  the  total  space  s,  time  in  seconds  /,  and  acquired 
velocity  v,  we  have,  from  the  above  laws, 

f:  v::  I  it. 


f/:.*:*/*:* 


27.  Applications  of  the  Formulae. — 

1.  A  body  moves  from  rest  during  6  seconds  with  acceleration 
f  =  20  ft. ;  what  space  does  it  pass  over,  and  what  velocity  does 
it  acquire  ?  Ans.  s  =  360  ft.,  v  =  120  ft  per  sec. 


16  MECHANICS. 

2.  A  train  starts  from  rest  and  moves  with  acceleration  of 
2  ft.  per  second,  and  finally  acquires  a  speed  of  60  miles  per  hour. 
or  88  ft.  per  second.     How  long  did  it  take  to  acquire  this  speed, 
and  what  distance  was  run  in  so  doing  ?    Here  /  =  2,  and  v  =  88. 

Ans.  44  sec. ;  space  =  1936  ft. 

The  mean  value  of  the  acceleration  due  to  the  action  of  gravity 
is  usually  given  as  32.2  ft.  per  second,  and  is  generally  represented 
by  g.  If  we  substitute  g  for  /  in  the  general  formulae,  when 

treating  of  falling  bodies,  we  will  have  t—  -,  v  =  g  t,  s  =  £  g  tz, 

3.  A  body  falls  from  rest  and  reaches  a  point  257.6  ft.  below. 
How  long  time  was  it  falling  and  what  velocity  did  it  acquire  ? 

Ans.  t  =  4  sec ;  v  =  128.8  ft.  per  sec. 

4.  A  body  moves  from  rest  during  4  seconds  and  acquires  a 
velocity  of  300  ft.    What  was  the  acceleration,  and  what  space 
was  passed  over  ?  Ans.  f  =  75  ft.  per  sec. ;  s  =  600  ft. 

5.  A  body  moves  from  rest  during  3£  seconds  and  passes  over 
147  feet.    What  was  the  acceleration,  and  what  the  final  velocity  ? 

Ans.  f  =  24  ft.  per  sec. ;  v  =  84  ft.  per  sec. 

6.  A  train  after  running  \  mile  has  acquired  a  velocity  of 
30  miles  an  hour.    What  was  the  acceleration,  and  how  long  had 
it  been  moving?      Ans.  f  =  J  mile  per  minute  ;  t  =  2  minutes. 

28.  Uniform  and  Uniformly  Varied  Motion  Combined. 

— Thus  far  we  have  assumed  the  body  to  start  from  rest.  If  the 
condition  be  changed  and  the  body  be  considered  as  having  a 
uniform  motion  at  the  time  the  action  of  the  constant  force 
begins,  we  have  merely  to  combine  the  formula  for  that  motion 
with  that  of  uniformly  accelerated  motion  already  used.  Thus, 
if  a  body  is  thrown  downward  with  a  force  which  gives  it  a  velocity 
of  40  ft.  per  second,  how  far  will  it  fall  in  4  seconds,  and  what 
velocity  will  it  have  at  the  end  of  that  time  ?  Under  the  action 
of  the  downward  impulse  alone,  it  would  move  over  4  x  40  ft.  = 
160  ft.  Under  the  action  of  gravity  it  would  move  over  16.1  x  4* 
=  257.6 ;  combining  these  two  eifects,  we  have  417.6  ft.  as  the 
total  distance  passed  over  in  the  given  time.  Designating  the 
velocity  due  to  the  impulse,  usually  called  the  "initial  velocity/* 
by  v,  we  have  total  space  s  =  v  t  -f  £  g  t2 ;  and  also  final  velocity 
v'  =  v  +  g  t  =  40  -j-  128.8  =  168.8  ft.  per  second. 

29.  Uniformly  Retarded   Motion. — In  like  manner  we 
can  determine  the  results  when  a  constant  force  acts  to  retard 


VARIABLE    MOTION.  —  CONSTANT    FORCES.          17 

velocity  already  imparted,  by  merely  taking  the  difference  of  the 
two  effects. 

A  body  receives  an  impulse  of  100  ft.  per  second,  and  is  retarded 
by  a  constant  force  whose  acceleration  is  10  ft.  per  second  ;  how 
far  will  the  body  move  in  5  seconds  ?  We  now  have  s  =  v  t  — 
s  =  100  x  5  —  5  x  25  =  375. 


30.  Applications  of  Formulae  for  the  Fall  of  Bodies.— 

1.  A  body  falls  6  seconds  ;  what  space  does  it  pass  over,  and 
what  velocity  does  it  acquire  ? 

Ans.  s  =  579.6  ft.,  v  =  193.2  ft.  per  sec. 

2.  How  far  must  a  body  fall  to  acquire  a  velocity  of  50  feet 
per  second,  and  how  long  will  it  be  in  falling  ? 

Ans.  s  =  38.8  ft.,  t  =  1.55  sec. 

3.  A  body  fell  from  the  top  of  a  tower  150  feet  high  ;  how  lony 
was  it  in  falling,  and  what  velocity  did  it  have  at  the  bottom  ? 

Ans.  t  =  3.05  sec.,  v  =  98.2  ft. 

4.  If  a  ball  be  thrown  upward  with  a  velocity  of  100  feet  per 
second,  what  height  will  it  reach  ?  Ans.  155.3  ft. 

5.  Suppose  a  body  to  fall  during  3  seconds,  and  then  to  move 
uniformly  during  2  seconds  more,  with  the  velocity  acquired  : 
what  is  the  whole  distance  passed  over  ? 

The  space  fallen  through  is  16.1  x  9  =  144.9  feet.  The  velo- 
city acquired  is  32.2  x  3  =  96.6  feet.  The  space  described  uni- 
formly is  96.6  x  2  =  193.2  feet.  Therefore  the  whole  space  is 
144.9  -f  193.2  =  338.1  feet. 

6.  A  ball  fired  perpendicularly  upward  was  gone  10  seconds, 
when  it  returned  to  the  same  place  ;  how  high  did  it  rise,  and  with 
what  velocity  was  it  projected  ?     Ans.  s  =  402.5  ft.,  v  =  161  ft. 

31.  Space  in  any  Given  Second  or  Seconds  of  Fall.  — 

If  it  be  required  to  find  how  far  a  body  will  move  during  any 
specified  unit  or  units  of  time,  proceed  thus  :  suppose  it  to  be 
required  to  determine  how  far  the  body  will  move  during  the 
7th  second  ;  for  the  whole  7  seconds, 

s  =  If*  =i/x  72; 
for  six  seconds,          s'  =  %ft'*  =  J-  /  x  62  ; 

s-s'=  \f  (f  -  **)  =  i/  (72  -  62) 

Suppose  we  are  required  to  determine  the  space  described 
during  the  last  three  seconds  : 

s  =  tfP  =i/x  72; 


18  MECHANICS. 

1.  How  far  does  a  body  move  in  the  14th  second  of  its  fall  ? 

Ans.  434.7  ft. 

2.  A  body  had  been  falling  2  minutes  ;  how  far  did  it  move 
in  the  last  second  ?  Ans.  3847.9  ft. 

3.  What  space  was  described  in  the  last  two  seconds  by  a  body 
which  had  fallen  300  feet  ?  Ans.  214.1  ft. 

4.  A  body  had  been  falling  8J  seconds  ;  how  far  did  it  descend 
in  the  next  second  ?  Ans.  289.8  ft. 

32.  Calculation  for  Projection  Upward  or  Downward.— 

A  body  projected  downward  describes  t  v  feet  by  the  force  of  pro- 
jection, and  -|  g  t2  feet  by  the  force  of  gravity  (Art.  28).  A  body 
projected  upward  describes  t  v  by  the  force  of  projection  ;  but  this 
is  diminished  by  £  g  fi,  which  gravity  would  cause  it  to  describe 
in  the  same  time  (Art  29).  Therefore  the  formula  for  space  de- 
scribed by  a  body  projected  downward  is  s  =  t  v  +  $  g  t* ;  by  a 
body  projected  upward,  the  formula  is  s  =  t  v  —  -J  g  tz. 

1.  A  body  is  projected  downward  with  a  velocity  of  30  feet  in 
a  second  ;  lioio  far  will  it  fall  in  4  seconds?  Ans.  377.6  ft. 

2.  A  body  is  projected  upward  with  a  velocity  of  120  feet  in  a 
second  ;  hoio  far  will  it  rise  in  3  seconds?  Ans.  215.1  ft. 

3.  Suppose  at  the  same  instant  that  a  body  begins  to  fall  from 
rest  from  the  point  D  (Fig.  4),  another  body  is  projected 
upward  from  B  with  a  velocity  which  would  carry  it  to  FIG.  4. 
A ;   it  is   required   to   find  the   point  where  they  would 
meet. 

Let  C  be  the  point  where  the  bodies  would  meet ;  and 
let  A  B  =  «,  B  D  =  b,  D  C  =  x\  then  will  A  D  =  a  —  b, 

Now  the  time  of  descending  through  D  C  =  I —  1   ;  and 
the  time  of  ascending  through  B  C  (=  time  down  A  B  — 
time  down  A  C)  =  ( — J   —  ( — - —         — -j    ;  but  the  time       -^ 
down  D  C  must  be  equal  to  the  time  up  B  G\  hence  we  have 


or  = 


.'.  (a  —  b-\-x)^  =  a?  —  x*9  and  a  —  b-\-x  =  a-\-x  —  2  (ax)^  • 

72 

.'.  2  (a  x)^  =  by  or  4  a  x  =  &*,  and  x  =  -  —  • 

4  Ctr 

4.  Suppose  a  body  to  have  fallen  from  A  to  B  (Fig.  5),  when 
another  body  begins  to  fall  from  rest  at  D  ;  how  far  will  the  latter 
body  fall  before  it  is  overtaken  by  the  former  ? 


FALL    OF    BODIES.  19 

Let  G  be  the  point  where  one  body  overtakes  the  other,  FIG.  5. 
=  l,DC  =  x;  then  A  C=a 


Now  time  down  D  C  =    —  )  ,  and  time  down  B  C  =  time 


, 
down  AC  —  time  down  A  B 


_  (™V 

\T/' 

2  (a  +  t  +  z)\i      {2  a\i 

T    '  ~  (-Jt  ' 

but  at  the  moment  when  the  lower  body  is  overtaken,  time 
down  D  C.  —  time  down  B  C,  or 

/2 ar\i__  /2  (a  +  b  +  s)\i      /2«\i 
VT/  :~V         (or         /    "VJ  ; 


.D 


33.  Questions  on  Falling  Bodies. — 

1.  The  momentum  of  a  meteoric  stone  at  the  instant  of     •   ^ 
striking  the  earth  was  estimated  at  18435  foot-pounds,  and 

it  had  been  falling  10  seconds ;  from  what  height  did  it  fall,  and 
what  was  its  weight  ?  Ans.  1610  ft.;  57.2  Ibs. 

2.  An  archer  wishing  to  know  the  height  of  a  tower,  found 
that  an  arrow  sent  to  the  top  of  it  occupied  8  seconds  in  going  and 
returning  ;  what  was  the  height  of  the  tower?       Ans.  257.6  ft. 

3.  In  what  time  would  a  man  fall  from  a  balloon  three  mijjes 
high,  and  what  velocity  would  he  acquire  ? 

Ans.  t  =  31.4  sec.;  v  =  1011.1  ft. 

4.  A  body  having  fallen  for  3£  seconds,  was  afterwards  observed 
to  move  with  the  velocity  which  it  had  acquired  for  2$  seconds 
more  ;  what  was  the  whole  space  described  by  the  body  ? 

Ans.  478.9  ft.,  very  nearly. 

5.  Through  what  space  would  the  aeronaut  (in  Question  3) 
fall  during  the  last  second  ?  .  Ans.  995  ft. 

6.  A  body  has  fallen  from  the  top  of  a  tower  340  feet  high  ; 
"what  was  the  space  described  by  it  in  the  last  three  seconds  ? 

Ans.  299.5  ft. 

7.  Suppose  a  body  be  projected  downward  with  a  velocity  of 
18  feet  in  a  second  ;  how  far  will  it  descend  in  15  seconds  ? 

Ans.  3892.5ft. 

8.  A  body  is  projected  upward  with  a  velocity  of  65  feet  in  a 
second  ;  how  far  will  it  rise  in  two  seconds  ?  Ans.  65.6  ft. 

9.  With  what  velocity  must  a  stone  be  projected  into  a  well 
450  feet  deep,  that  it  may  arrive  at  the  bottom  in  four  seconds  ? 

Ans.  48.1  ft.  in  a  second. 

10.  The  space  described  in  the  fourth  second  of  fall  was  to  the 
space  described  in  the  last  second  except  four,  as  1  :  3  ;  what 
the  whole  space  described  by  the  body  ?  Ans.  3622.5  ft. 


20  MECHANICS. 

11.  A  staging  is  at  the  height  of  84  ft.  above  the  earth.     A 
ball  thrown  upward  from  the  earth,  after  an  absence  of  7  seconds, 
fell  on  the  staging  ;  what  was  the  velocity  of  projection  ? 

Ans.  124.7  ft.  per  second. 

12.  A  body  is  projected  upward  with  a  velocity  of  483  feet  in 
a  second  ;  in  what  time  will  it  rise  to  a  height  of  1610  feet? 

Ans.  t  =  3.8  sec.,  or  26.2  sec.. 

13.  From  a  point  386.4  feet  above  the  earth  a  body  is  pro- 
jected upward  with  a  velocity  of  161  feet  in  a  second  ;  in  what 
time  will  it  reach  the  surface  of  the  earth,  and  -with  what  velocity 
will  it  strike?  Ans.  t  —  12  sec.,  v  —  225.4  ft. 

14.  A  body  is  projected  upward  with  a  velocity  of  64.4  feet  in 
a  second ;  how  far  above  the  point  of  projection  will  it  be  at  the 
end  of  4  seconds  ?  Ans.  0  ft. 

15.  A  body  is  projected  upward  with  a  velocity  of  128.8  feet 
in  a  second  ;  where  will  it  be  at  the  end  of  10  seconds  ? 

Ans.  322  ft.  below  the  point  of  projection. 

34.  Determination  of  the  Acceleration  due  to  a  Given 
Force. — Thus  far  forces  have  been  represented  by  their  accelera- 
tions/., but  we  have  no  definite  idea  as  to  how  much  acceleration 
a  force  of  9  Ibs.  would  impress  upon  a  mass  of  13  Ibs.  By  a  force 
of  9  Ibs.  is  meant  a  constant  pressure  or  tension  which  would  be 
exactly  balanced  by  a  weight  of  9  Ibs.  or  by  a  spring  which  would 
sustain  such  weight. 

If  a  force  will  produce, an  acceleration  of  /"ft.  per  sec.  in  a 
mass  m,  the  same  force  would  produce  an  acceleration  of  only 

m 

J /ft.  per  sec.  in  a  mass  2  m,  or  2 /ft.  per  sec.  in  a  mass  —  • 

2o 

If  a  force  will  produce  an  acceleration  /ft.  per  sec.  in  a  mass 
m,  twice  that  force  would  produce  an  acceleration  of  2  /  ft.  per 
sec.  in  the  same  mass. 

Now  a  force  of  9  Ibs.  acting  upon  a  mass  of  9  Ibs.  produces  an 
acceleration  of  32.2  ft.  per  sec.  in  the  case  of  a  falling  body.  But 
tliis  same  force,  if  acting  upon  twice  as  much  matter,  or  a  mass 

32  2 
of  18  Ibs.  could  impress  only  one-half  as  much  acceleration  or  -  ^-  • 

A 

From  this  example,  we  see  that  the  acceleration  /,  due  to  a  force, 
is  such  part  of  the  acceleration  which  gravity  would  impress  as 

E7 

the  force  .Pin  Ibs.,  is  of  the  mass  Win  Ibs. ;  hence /=  ^  x  32.2, 

and  this  value  of  /must  be  substituted  in  the  formulae  of  Art.  26 
when  applied  to  problems  like  the  following  : 

If  a  body  whose  weight  is  40  Ibs.  is  moved  horizontally  by  a 
constant  force  of  10  Ibs.,  how  far  would  it  move  during  5  seconds, 


ATWOOD'S    MACHINE. 


and  what  velocity  would  it  acquire,  no  friction  nor  other  resist- 
ances being  considered  ? 

The  acceleration  of  the  mass  40  Ibs.,  due  to  the  force  10  Ibs.,  is 

/  —  -~  x  32.2  =  8.05  ft.  per  sec.  FlG  6 

Hence, 


-y  — 


=  -    -  X  52  =  100. 6  ft.,  and 


v  =ft  =  8.05  x  5  =  40.2  ft.  per  sec. 

35.  Atwood's  Machine. — All  the 

facts  of  uniformly  accelerated  or  re- 
tarded motion  may  be  illustrated  with 
sufficient  accuracy  by  Atwood's  ma- 
chine, since  in  accordance  with  what 
has  just  been  shown,  we  can  render  the 
motion  of  the  parts  as  slow  as  we 
choose,  and  therefore  bring  the  velo- 
cities of  the  moving  masses  within  the 
limits  of  observation.  From  the  base 
of  the  instrument,  which  is  furnished 
with  leveling  screws,  rises  a  substantial 
pillar,  about  seven  feet  high,  supporting 
a,  small  table  upon  the  top  (Fig.  6). 

Above  the  table  is  a  grooved  wheel, 
delicately  suspended  on  friction-wheels, 
and  protected  from  dust  by  a  glass  case. 
Two  equal  poises,  M  and  M',  are  at- 
tached to  the  ends  of  a  fine  cord,  which 
passes  over  the  groove  of  the  wheel.  As 
gravity  exerts  equal  forces  on  M  and 
M',  they  are  in  equilibrium.  To  set 
them  in  motion,  a  small  bar  m  is  placed 
on  J/,  which  will  immediately  begin  to 
descend,  and  M'  to  rise.  But  this  mo- 
tion will  be  slower  than  in  falling  freely, 
because  the  force  which  gravity  exerts 
on  the  bar  must  be  communicated  to 
the  poises,  and  also  to  the  revolving 
wheel  over  which  the  cord  passes.  By 
increasing  the  poises  M,  M't  and  dimin- 
ishing the  bar  m,  the  motion  may  be 

made  as  slow  as  we  please.  H  is  a  simple  clock  attached  to  the 
pillar  for  measuring  seconds,  and  for  dropping  the  poise  M  at  the 
beginning  of  a  vibration  of  the  pendulum.  Q  is  a  scale  of  inches 


22  MECHANICS. 

extending  from  the  base  to  the  table.  The  stage  A  may  be 
clamped  to  any  part  of  the  scale,  in  order  to  stop  the  poise  M  in 
its  descent,  as  represented  at  C.  The  ring  B,  which  is  large 
enough  to  allow  the  poise,  but  not  the  bar,  to  pass  through 
it,  is  also  clamped  to  the  scale  wherever  the  acceleration  is  to 
cease. 

Let  M  be  raised  to  the  top,  and  held  in  place  by  a  support,  and 
then  let  the  pendulum  be  set  vibrating.  When  the  index  passes 
the  zero  point,  the  clock  causes  the  support  to  drop  away,  and  the 
poise  descends.  The  pendulum  shows  how  many  seconds  elapse 
before  the  bar  is  arrested  by  the  .ring,  and  how  many  more  before 
the  poise  strikes  the  stage.  From  the  top  to  the  ring  the  motion 
is  accelerated  by  the  constant  fraction  of  gravity  acting  on  it ; 
from  the  ring  to  the  stage  the  poise  moves  uniformly  with  the 
acquired  velocity.  Moreover,  the  resistance  of  the  air  is  so  much 
diminished  when  the  motion  is  slow,  that  a  good  degree  of  cor- 
respondence is  found  to  exist  between  the  experiments  and  the 
results  of  calculation. 

If  we  disregard  the  mass  of  the  wheel  as  not  sensibly  affect- 
ing the  results,  which  we  may  do  in  practice  if  the  weights  are 
heavy  as  compared  with  it,  we  may  illustrate  the  action  of  the 
machine  by  the .  following  case  :  Suppose  the  weights  to  be  7^ 
ounces  each.  They  will  balance  and  no  motion  will  result.  Now 
lay  a  weight  of  one  ounce  upon  one  of  them,  and,  equilibrium 
being  destroyed,  motion  will  ensue.  In  this  we  have  a  force  of 
one  ounce  moving  a  mass  of  sixteen  ounces,  and  the  resulting 
velocity  will  be  determined  by  reference  to  paragraph  34,  to  be 
about  two  feet  at  the  end  of  the  1st  second,  a  velocity  readily 
noted. 

36.  Work, — If  a  force  moves  a  body  over  a  distance  b  ft. 
against  a  resistance  of  a  Ibs.,  work  is  said  to  be  done  ;  hence  the 
measure  of  the  work  done  involves  the  resistance  overcome  and 
the  distance  moved.  The  unit  of  work  which  will  be  used  here- 
after, is  the  work  done  in  raising  a  weight  of  1  Ib.  through  a 
height  of  one  foot,  and  this  unit  is  called  a  "foot-pound." 

If  10  Ibs.  be  raised  through  1  ft.,  ten  foot-pounds  is  the  work 
done,  or  if  1  Ib.  be  raised  through  ten  feet,  the  work  is  ten  foot- 
pounds also. 

The  product  of  the  resistance  in  pounds  ~by  the  distance  in  feet 
through  which  the  body  is  moved  against  the  resistance,  is  the 
measure  of  the  work  done. 

Ex.  1.  A  body  is  moved  on  a  horizontal  plane  against  a  resist- 
ance of  50  Ibs.,  due  to  friction  ;  how  much  work  is  done  in  moving 
it  over  100  ft.  ?  Ans.  5000  foot-pounds. 


LIVING   FORCE.  23 

Ex.  2.  A  train  weighing  100  tons  moves  30  miles,  resistance 
being  8  Ibs.  per  ton  ;  how  much  work  is  expended  in  this  case  ? 

Am.  126,720,000  foot-pounds. 

37.  Living  Force. — In  order  to  give  motion  to  a  body  work 
must  be  done  upon  it ;  if  after  having  acquired  a  certain  velocity 
resistance  be  opposed  to  its  motion,  it  will  in  its  turn  do  work 
in  overcoming  such  resistance.  In  order  to  determine  the  work 
which  a  moving  body  is  capable  of  doing,  we  have  merely  to  deter- 
mine the  height  from  which  the  body  must  fall  to  acquire  the 
given  velocity,  and  the  product  of  such  height  in  feet  by  the 
weight  of  the  body  in  pounds  will  be  the  work  which  it  can  do 
before  coming  to  rest;  for  the  work  it  can  do  is  only  that  which 
has  been  stored  up  in  it,  which  we  may  call  "  accumulated  work ;" 
that  is  to  say  the  work  done  in  raising  it  to  the  height  determined, 
or  what  is  the  same  thing,  the  work  done  upon  it  by  gravity  while 
falling  through  this  height. 

Let  u  =  number  of  units  of  work  accumulated  in  a  body 
whose  weight  is  w,  and  velocity  v  ft.  per  second.  Put  s  for  the 
height  in  feet  from  which  the  body  must  fall  to  acquire  the  given 

velocity  ;  then  u  =  w  s,  but  by  the  laws  of  falling  bodies,  s  =  = — 

t/ 

_  w  v2 

•   •      tv    "~7x         * 


The  expression  -    -  is  called  the  living  force  of  the  body  ;  there- 

%j 

fore,  one-half  the  living  force  is  the  measure  of  the  ivork  accumu- 
lated in  a  body. 

The  force  of  gravity  g  acts  upon  every  particle  of  a  body  alike  ; 
therefore  the  greater  the  number  of  these  particles,  or  the  greater 
its  mass,  the  greater  will  be  its  weight,  or  w  =  g  g  from  which 

we  have  q  =  — ,  which,  in  the  expression  for  living  force,  gives 

i/ 

us  -   -  =  q  v*,  while  momentum  =  q  v.      To  compare  these, 

suppose  a  ball  weighing  one  pound  to  move  with  the  velocity  of 
2000  feet,  and  another  ball  weighing  two  pounds  to  move  with 
the  velocity  of  1000  feet,  then  the  momentum  ( g  v)  of  the  first 
equals  that  of  the  second.  But  the  living  force  ( q  v2)  of  the  first  is 
twice  as  great  as  that  of  the  second  ;  for  1  x  20002 :  2  x  10002 : :  2  : 1. 
To  give  a  missile  greater  velocity  is  more  advantageous  than  to 
increase  its  mass.  A  40-pound  ball  with  1400  feet  velocity,  is 
7  times  more  efficient  in  penetrating  the  walls  of  forts  and  the 
hulks  of  ships  than  a  280-pound  ball  with  200  feet  velocity, 
though  the  momentum  is  the  same  in  each  case. 


24  MECHANICS. 

38.  Applications  of  Living  Force  to  Work.— 

1.  A  body  weighing  100  Ibs.  is  moving  at  the  rate  of  50  ft. 
per  second.     What  space  will  it  move  through  against  a  resistance 
of  5  Ibs.  before  it  comes  to  rest  ? 

Let  s  =  the  space  in  feet ;  then  as  5  Ibs.  has  been  overcome 
through  distance  of  s  feet,  the  work  done  is  5  s  foot-pounds  =  u  ; 

w  v2 
but  u  =  -£ —  ; 

100x502    , 

.-.  5  s  =  -  — ^7-7*  from  which  we  find 
/c  X  o</.*6 

8  =  7764-  ft. 

2.  Kequired  the  time  before  the  body  will  come  to  rest,  in  the 
last  problem. 

The  resistance,  5  Ibs.,  is  a  constant  force,  and  the  body  will 
move  with  uniformly  retarded  motion.  If  it  has  a  velocity  of 
50  ft.  per  second  at  the  start  and  0  at  the  end  of  the  time,  its 

776 

average  velocity  is  25  ft.  per  second ;  hence,  — -  will  give  the 

/*o 

number  of  seconds  required,  or  t  =  31.04  sec. 

3.  A  railway  train  weighing  100  tons  has  a  velocity  of  30  ft. 
per  second,  resistances  being  8  Ibs.  per  ton  ;  what  distance  will 
the  train  move,  on  a  level,  after  steam  is  shut  off  ? 

Ans.  3494—  ft. 

4.  Two  men  are  pulling  a  boat  ashore  by  a  rope,  one  at  each 
end,  A  being  in  the  boat  and  B  on  the  shore  ;  how  will  the  time 
of  bringing  the  boat  ashore  compare  with  the  time  in  which  A 
would  pull  it  ashore  alone,  were  the  other  end  of  the  rope  fixed 
to  an  immovable  post  ? 

5.  Suppose  the  rope  to  pass  from  A  in  one  boat  to  B  in 
another  equal  boat ;  how  fast  will  £'a  boat  move  ?  will  A's  boat; 
have  the  same  velocity  as  when  B  was  on  the  shore  ? 

39.  Measure  of  Force. — In  Art.  12  force  is  said  to  be 
measured  by  momentum,  or  f  oc  q  v ;  and  in  Art.  37  it  is  said  to  be 
measured  by  the  work  performed,  or  /  ex  q  v2.     But  these  state- 
ments are  not  to  be  considered  as  inconsistent  with  each  other  ; 
for  in  the  first  case,  force  has  reference  to  inertia  ;  in  the  second 
case  it  has  reference  to  work.     When  a  force  acts  on  a  body  that 
is  free  to  move  without  obstruction  (which  is,  however,  only  a 
supposable  case),  the  effect  is  perpetual ;  the  body  will  move  on 
uniformly  forever.     If  the  force  had  been  greater,  the  velocity 
would  have  been  greater  in  the  same  ratio.     But  when  resistances 
oppose  (as  is  always  true  in  practice),  then  the  force  is  expended 
in  overcoming  them,  and  this  is  the  ivork  to  be  performed  ;  and 
if  the  force  ceases  to  operate,  the  motion  will  at  length  cease  also ; 


THE  PARALLELOGRAM  OF  FORCES.       25 

but,  as  has  been  shown,  the  space  passed  over,  and  therefore  the 
work  performed,  will  vary  as  the  square  of  the  velocity. 

When  force  is  employed  to  perform  work,  it  is  by  some  writers 
called  energy,  to  distinguish  it  from  force  as  used  ill  producing 
momentum. 


CHAPTER   III. 

COMPOSITION    AND    RESOLUTION    OF    MOTION. 

40.  Motion  by  Two  or  More  Forces. — Motion  produced 
by  a  single  force,  either  impulsive  or  continued,  has  been  already 
considered.     But  motion  is  more  generally  caused  by  several  forces 
acting  in  different  directions. 

When  two  or  more  forces  act  at  once  on  a  body,  each  force  is 
called  a  component,  and  the  joint  effect  is  called  the  resultant. 
Forces  may  be  represented  by  the  straight  lines  along  which  they 
would  move  a  body  in  a  given  time  ;  the  lines  represent  the  forces 
in  two  particulars,  the  directions  in  which  they  act  and  their  rela- 
tive magnitudes.  Whenever  an  arrow-head  is  placed  on  a  line,  it 
.shows  in  which  of  the  two  directions  along  that  line  the  force  acts. 

41.  The   Parallelogram  of  Forces. — This  is  the  name 
given  to  the  relation  which  exists  between  any  two  components 
and  their  resultant,  and  is  stated  as  follows  : 

If  two  forces  acting  at  once  on  a  body  are  represented  by  the 
adjacent  sides  of  a  parallelogram,  their  resultant  is  expressed  by 
the  diagonal  ivhich  passes  through  the  intersection  of  those  sides. 

Suppose  that  a  body  situated  at  A  (Fig.  7)  receives  an  impulse 
which,  acting  alone,  would  carry  it 
over  A  B  in  a  given  time,  and  an-  IG* 

other  which  would  carry  it  over  A  C 
in  the  same  length  of  time.  If  both 
impulses  are  given  at  the  same  in- 
stant, the  body  describes  A  D  in  the 
same  time  as  A  B  by  the  first  force, 
or  A  C  by  the  second,  and  the  motion  in  A  D  is  uniform. 

This  is  an  instance  of  the  coexistence  of  motions,  stated  in  the 
second  law  of  motion  (Art.  13).  For  the  body,  in  passing  directly 
from  A  to  D,  is  making  progress  in  the  direction  A  C  as  rapidly 
as  though  the  force  A  B  did  not  exist ;  and  at  the  same  time  it 
advances  in  the  direction  A  B  as  fast  as  though  that  were  the 


26  MECHANICS. 

only  force.  When  the  body  reaches  7),  it  is  as  far  from  the  line 
A  B  as  if  it  had  passed  over  A  G\  it  is  also  as  far  from  the  line 
A  C  as  if  it  had  gone  over  A  B.  Thus  it  appears  that  both  mo- 
tions A  B  and  A  G  fully  coexist  in  the  progress  of  the  body  along 
the  diagonal  A  D.  That  the  motion  is  uniform  in  the  diagonal 
is  evident  from  the  law  of  inertia ;  for  the  body  is  not  acted  on 
after  it  leaves  A. 

It  is  evident  that  a  single  force  might  produce  the  same  effect ; 
that  force  would  be  represented,  both  in  direction  and  magnitude, 
by  the  line  A  D.  The  force  A  D  is  said  to  be  equivalent  to  the 
two  forces  A  B  and  A  C. 

42.  Velocities  Represented. — The  lines  A  B  and  A  C  are 
described  by  the  components  separately,  and  the  line  A  D  by  their 
joint  action,  in  the  same  length  of  time.     Hence  the  velocities  in 
those  lines  are  as  the  lines  themselves.     In  the  parallelogram  of 
forces,  therefore,  two  adjacent  sides  and  the  diagonal  between 
them  represent — 

1st.  The  relative  directions  of  the  components  and  resultant ; 

2d.  Their  relative  magnitudes  ;  and 

3d.  The  relative  velocities  with  which  the  lines  are  described. 

43.  The  Triangle  of  Forces. — For  purposes  of  calculation, 
it  is  more  convenient  to  represent  two  components  and  their  re- 
sultant by  the  sides  of  a  triangle,  than  by  the  sides  and  diagonal 
of  a  parallelogram.     In  Fig.  7,  C  D9  which  is  equal  and  parallel  to 
A  B,  may  represent  in  direction  and  magnitude  the  same  force 
which  A  B  represents.     Therefore,  the  components  are  A  G  and 
G  D,  while  their  resultant  is  A  D ;  and  the  angle  G  in  the  triangle 
is  the  supplement  of  GAB,  the  angle  between  the  components. 
Care  should  be  taken  to  construct  the  triangle  so  that  the  sides 
representing  the  components  may  be  taken  in  succession  in  the 
directions  of  the  forces,  as  A  G,  G  D  ;  then  A  D  correctly  repre- 
sents their  resultant.    But,  although  A  G  and  A  B  represent  the 
components,  the  third  side,  C  B,  of  the  triangle  A  C  B,  does  not 
represent  their  resultant,  since  A  G  and  A  B  cannot  be  taken  suc- 
cessively in  the  direction  of  the  forces.     It  is  necessary  to  go  back 
to  A  in  order  to  trace  the  line  A  B.     It  should  be  observed,  that 
though  C  D  represents  the  magnitude  and  direction  of  the  compo- 
nent, it  is  not  in  the  line  of  its  action,  because  both  forces  act 
through  the  same  point  A. 

Three  forces  produce  equilibrium  when  they  may  be  represented 
in  direction  and  magnitude  by  the  three  sides  of  a  triangle  taken 
in  order. 


TRIANGLE   OF    FORCES. 


FIG.  8. 


For,  when  three  forces  are  in  equilibrium,  one  of  them  must 
be  equal  to,  and  opposite  to,  the  re- 
sultant of  both  the  others.  But 
the  forces  A  C  and  A  B  (Fig.  8) 
produce  the  resultant  A  D  ;  there- 
fore the  equal  and  opposite  force 
D  A,  since  it  is  in  equilibrium  with 
A  D,  is  also  in  equilibrium  with  A  C 
and  A  B,  or  A  C  and  C  D.  Hence 

the  three  forces  A  C,  C  D,  and  D  A,  taken  in  order  around  the 
figure,  produce  equilibrium. 

44.  The  Forces  Represented  Trigonometrically. — Since 
the  sides  of  a  triangle  are  proportional  to  the  sines  of  the  opposite 
angles,  these  sines  may  also  represent  two  components  and  their 
resultant.    Thus,  the  sine  of  C  A  D  corresponds  to  the  component 
A  B  (=  C  D) ;  the  sine  of  C  D  A  (=  D  A  B)  corresponds  to  the 
component  A  (7;  and  the  sine  of  C(=  sine  of  C  A  B)  corresponds 
to  the  resultant  A  D.    Each  of  the  three  forces  is  therefore  repre- 
sented by  the  sine  of  the  angle  between  the  other  two. 

45.  Greatest  and  Least  Values  of  the  Resultant. — A 

change  in  the  angle  between  the  components  alters  the  value  of 
the  resultant ;  as  the  angle  increases  from  0°  to  180°,  the  resultant 
diminishes  from  the  sum  of 
the  components  to  their  differ- 
ence.   In  Fig.  9,  let  C  A  B 
and  D  A  B  be  two  different 
angles  between  the  same  com- 
ponents A  C  (or  A  D)  and 
A  B.     As  C  A  B  is  less  than 
D  A  Bf  its  supplement  A  B  F 
is  greater  than  ABE,  the 
supplement  of  D  A  By  there- 
fore A  F  is  greater  than  A  E.     When  the  angle  C  A  B  is  dimin- 
ished to  0°,  the  sides  A  B,  B  F,  become  one  straight  line,  and  A  F 
equals  their  sum  ;  when  D  A  B  is  enlarged  to  180°,  E  falls  on 
A  B,  and  A  E  equals  the  difference  of  A  B  and  A  C.    Between 
the  sum  and  difference  of  the  components,   the  resultant  may 
have  all  possible  values. 

Two  forces  produce  equilibrium  when  they  are  equal  and  act 
upon  the  same  point  in  opposite  directions. 

Since  two  forces  produce  the  least  resultant  when  they  act  at 
an  angle  of  180°  with  each  other,  and  the  resultant  then  equals 
the  difference  of  the  forces,  if  the  forces  are  equal,  their  difference 


9. 


MECHANICS. 


FIG.  10. 


FIG.  11. 


is  zero,  and  the  resultant  vanishes  ;  that  is,  the  two  forces  pro- 
duce equilibrium. 

46.  The  Polygon  of  Forces. — All  the  sides  of  a  polygon 
except  one  may  represent  so  many  forces  acting  at  the  same  time 
on  a  body,  and  the  remaining  side  will  represent  their  resultant. 
In  Fig.  10,  suppose  A  B,  A  C, 
and  A  D,  to  represent  three 
forces  acting  together  on  a  body 
at  A.  The  resultant  of  A  B  and 
A  C  is  represented  by  the  diag- 
onal A  E\  and  the  resultant 
of  A  E  and  A  D  by  the  diagonal 
A  F.  As  A  F  is  equivalent  to 
A  E  and  A  D,  and  A  E  is  equiv- 
alent to  A  B  and  A  C,  therefore 
A  F  is  equivalent  to  the  three,  A  B,  A  C,  and  A  D.  But  if  we 
substitute  B  E  f  or  A  C,  and  E  F  for  A  D,  then  the  three  compo- 
nents are  A  B,  B  E,  and  E  F,  three  sides  of  a  polygon,  and  the 
resultant  A  F  is  the  fourth  side  of  the  same  polygon. 

So,  in  Fig.  11,  A  B,  B  C,  C  D,  D  E,  and  E  F,  may  represent 
the  directions  and  relative  magni- 
tudes of  five  forces,  which  act  simul- 
taneously on  a  body  at  A.  The  re- 
sultant of  A  B  and  B  C  is  A  C ;  the 
resultant  of  A  C  and  C  D  is  A  D  ; 
the  resultant  of  A  D  and  D  E  is 
A  E ;  and  the  resultant  of  A  E  and 
E  F\s  A  Fj  which  last  is  therefore 
the  resultant  of  all  the  forces,  A  B, 
B  C,  C  D,  D  E,  and  E  F ;  the  com- 
ponents being  represented  by  five 
sides,  and  their  resultant  by  the  sixth  side,  of  a  polygon  of  six  sides. 

More  than  three  forces  in  one  plane  will  produce  equilibrium  when 
they  can  be  represented  by  the  sides  of  a  polygon  taken  in  order. 

Since  several  forces  acting  on  a  body,  are  represented  by  all 
the  sides  of  a  polygon  except 
one,  their  resultant  is  represented 
by  the  remaining  side.  Thus,  the 
resultant  of  the  forces  A  E,  B  C, 
C  D,  and  D  E  (Fig.  12),  is  A  E. 
Now,  the  force  E  A,  equal  and 
opposite  to  A  E,  since  it  would  be 
in  equilibrium  with  A  E,  is  there- 
fore in  equilibrium  with  all  the 


FIG.  12. 


CURVILINEAR    MOTION. 


FIG.  13. 


others.     Hence  the  forces  A  B,  B  C,  C  D,  D  E,  and  E  A,  taken 
in  order  around  the  figure,  are  in  equilibrium. 

47.  Curvilinear  Motion. — Since,  according  to  the  first  law 
of  motion,  a  moving  body  proceeds  in  a  straight  line,  if  no  force 
disturbs  it,  whenever  we  find  a  body  describing  a  curve,  it  is  cer- 
tain that  some  force  is  continually  deflecting  it  from  a  straight 
line.      Besides  the   original    impulse,   therefore,   which  gave  it 
motion  in  one  direction,  it  is  subject  to  the  action  of  a  continued 
force,  which  operates  in  another  direction.     A  familiar  example 
occurs  in  the  path  of  a  projectile.     Suppose  a  body  to  be  thrown 
from  P  (Fig.  13),  with  an  impulse  which  would  alone  carry  it  to 
N,  in  the  same  time  in  which  gravity  alone  would  carry  it  to  V+ 
Complete  the  parallelogram  P  Q  ; 

then,  as  both  motions  coexist  (2d 
law),  the  body  at  the  end  of  the 
time  will  be  found  at  Q.  Let  t  be 
the  time  of  describing  P  Nor  P  V\ 
and  let  t'  be  the  time  of  describing 
P  M  by  the  impulse,  or  P  L  by 
gravity.  Then,  at  the  end  of  the 
time  /',  the  body  will  be  at  0.  Now, 
as  P  N  is  described  uniformly, 
PN:PMr.t:  t'; .'.  P  N* :  PM*:: 

But  (Art.  25),  PV:PL::P: 
t'*',  .'.P  V:PL  r.PN*:PM*i  or  Q  V* :  0  Z2. 

Hence,  the  curve  is  such  that  P  Foe  Q  V* ;  that  is,  the  abscissa 
varies  as  the  square  of  the  ordinate,  which  is  a  property  of  the 
parabola.  P  0  Q  is  therefore  a  parabola,  one  of  whose  diameters 

is  P  V,  and  the  parameter  to  that  diameter  is  -py- • 

Owing  to  the  resistance  of  the  air,  the  curve  deviates  sensibly 
from  a  parabola,  especially  in  swift  motions. 

48.  Calculation  of  the  Resultant  of  Two  Impulsive 
Forces. — When  two  components  and  the  angle  between  them  are 
given,  the  resultant  may  be  found  both  in  direction  and  magni- 
tude by  trigonometry.     The  theorem  required  is  that  for  solving  a 
triangle,  when  two  sides  and  the  included  angle  are  given  ;  but 
the  included  angle  is  not  that  between  the  components,  but  its 
supplement  (Art.  43).    In  Fig.  7,  if  A  B  =  54,  and  A  C  =  22,  and 
GAB  —  75°,  then  A  G  D  is  the  triangle  for  solution,  in  which 
A   C  =  22,   C  D  =  54,  and  A    C  D  =  105°.     Performing  the 
calculation,  we  find  the  resultant  A  D  =  63.363,  and  the  angle 


30  MECHANICS. 

D  A  Ey  which  it  makes  with  the  greater  force,  =  19°  35'  4b  . 
This  method  will  apply  in  all  cases. 

1.  A  foot-ball  received  two   blows  at  the  same  instant,  one 
directly  east  at  the  rate  of  71  feet  per  second,  the  other  exactly 
northwest,  at  the  rate  of  48  feet  per  second ;  in  what  direction 
and  with  what  velocity  did  it  move  ? 

Am.  N.  47°  30'  52"  E.     Vel.  =50.253. 

The  process  is  of  course  abridged,  if  the  forces  act  at  a  right 
angle  with  each  other,  as  in  the  following  example  : 

2.  A  balloon  rises  1120  feet  in  one  minute,  and  in  the  same 
time  is  borne  by  the  wind  370  feet ;    what  angle  does  its  path 
make  with  the  vertical,  and  what  is  its  velocity  per  second  ? 

Ans.lS0  16' 53";  v  =  19.659. 

In  the  next  example,  one  component  and  the  angle  which  each 
component  makes  with  the  resultant,  are  given  to  find  the  result- 
ant and  the  other  component. 

3.  From  an  island  in  the  Straits  of  Sunda,  we  sailed  S.  E.  by 
S.  (33°  45')  at  the  rate  of  6  miles  an  hour ;  and  being  carried  by  a 
current,  which  was  running  toward  the  S.  W.  (making  an  angle 
with  the  meridian  of  64°  12 J'),  at  the  end  of  four  hours  we  came 
to  anchor  on  the  coast  of  Java,  and  found  the  said  island  bearing 
due  north ;  required  the  length  of  the  line  actually  described  by 
the  ship,  and  the  velocity  of  the  current  ? 

Ans.  s  =  26.4  miles. 

v  =  3. 7024  miles  per  hour. 

If  the  magnitudes  and  directions  of  any  number  of  forces  are 
given,  the  resultant  of  them  all  is  obtained  by  a  repetition  of  the 
same  process  as  for  two.  In  Fig.  11,  first  calculate  A  C,  and  the 
angle  A  0  B,  by  means  oiAB,B  0,  and  the  angle  B.  Subtracting 
A  C  B  from  B  C  D,  we  have  the  same  data  in  the  next  triangle, 
to  calculate  A  D,  and  thus  proceed  to  pIGi  14 

the  final  resultant,  A  F. 

As  it  is  immaterial  in  what  order 
the  components  are  introduced  into 
the  calculation,  it  will  diminish  labor, 
to  find  first  the  resultant  of  any  two 
equal  components,  or  any  two  which 
make  a  right  angle  with  each  other ; 
since  it  can  be  done  by  the  solution 
of  an  isosceles,  or  a  right-angled  tri- 
angle. 

4.  The  particle  A  (Fig.  14)  is 
urged  by  three  equal  forces  A  B, 
AC,  and  A  D  ;  the  angle  B  A  C  =  90°,  and  CA  D  =  45°  ; 


RESULTANT  OF  IMPULSIVE   FORCES. 


31 


what  is  the   direction  of  the  resultant,   and  how  many  times 
A  B  ?  Ans.  B  A  F  —  80°  16',  and 

A  F  :  A  B  :  :  y3  :  1. 

5.  Five  sailors  raise  a  weight  by  means  of  five  separate  ropes, 
in  the  same  plane,  connected  with  the  main  rope  that  is  fastened 
to  the  weight  in  the  manner  represented  in  Fig.  16.  B  pulls  at 
an  angle  with  A  of  20°;  C  with  B,  19°;  D  with  C,  21°  30';  and 
E  with  D,  25°.  A,  B,  and  C,  pull  with  equal  forces,  and  D  and 
E  with  forces  one-half  greater  ;  required  the  magnitude  and 
direction  of  the  resultant. 

Ans.  Its  angle  with  A  is  46°  33'  10".     Its  magnitude  is  5.1957 
times  the  force  of  A. 


FIG.  15. 


FIG.  16. 


If  the  polygon  0  A  B  G  D  E  (Fig.  15)  be  constructed  for  the 
above  case,  0  B  and  C  E  are  easily  calculated  in  the  isosceles  tri- 
angles 0  A  B  and  C  D  E,  after  which  0  C  and  then  0  E  are  to 
be  obtained  by  the  general  theorem. 

49.  The     Resultant    and    all  FIG.  17. 

Components,  except  one,  being 
given,  to  Find  that  one  Compo- 
nent.—If  A  B  (Fig.  17)  is  the  result- 
ant to  be  produced,  and  there  already 
exists  the  force  A  0,  a  second  force  -A 
can  be  found,  which  acting  jointly  with 
A  C,  will  produce  the  motion  required. 
Join  C  By  and  draw  A  D  equal  and  par- 
allel to  it,  then  A  D  is  the  force  re- 


32  MECHANICS. 

quired  ;  f or  A  B  is  equivalent  to  A  C  and  C  B.  Therefore  C  B 
has  the  magnitude  and  direction  of  the  required  force ;  A  D  is 
the  line  in  which  it  must  act. 

Again,  suppose  that  several  forces  act  on  A,  and  it  is  required 
to  find  the  force  which,  in  conjunction  with  them  all,  shall  pro- 
duce the  resultant  A  B.  Let  the  several  forces  be  combined  into 
one  resultant,  and  let  A  C  represent  that  resultant.  Then  A  D 
may  be  found  as  before. 

The  trigonometrical  process  for  finding  a  component  is  essen- 
tially the  same  as  for  finding  a  resultant. 

1.  A  ferry-boat  crosses  a  river  f  of  a  mile  broad  in  45  minutes, 
the  current  running  all  the  way  at  the  rate  of  3  miles  an  hour  ;  at 
what  angle  with  the  direct  course  must  the  boat  head  up  the  stream 
in  order  to  move  perpendicularly  across  ?  Ans.  71°  34'. 

2.  A  sloop  is  bound  from  the  mainland  of  Africa  to  an  island 
bearing  W.  by  N.  (78°  45')  distant  76  miles,  a  current  setting 
N.  N.  W.  (22°  30')  3  miles  an  hour ;  what  is  the  course  to  arrive 
at  the  island  in  the  shortest  time,  supposing  the  sloop  to  sail  at 
the  rate  of  6  knots  per  hour  ;  and  what  time  will  she  take  ? 

Ans.  Course,  S.  76°  41'  4"  W.    Time,  10  h.  40  m.  7  sec. 

3.  The  resultant  of  two  forces  is  10  ;  one  of  them  is  8,  and  the 
direction  of  the  other  is  inclined  to  the  resultant  at  an  angle  of  36°. 
Find  the  angle  between  the  two  forces. 

Ans.  47°  17'  5"  or  132°  42'  55". 

4.  A  ball  receives  two  impulses  :  one  of  which  would  carry  it 
N.  27  feet  per  second  ;  the  other  N.  60°  E.  with  the  same  velocity  ; 
what  third  impulse  must  be  conjoined  with  them,  to  make  the  ball 
go  E.  with  a  velocity  of  21  ft  ?     Ans.  S.  3°  22'  W.    v  =  40.57. 

50.  Resolution  of  Motion. — In  the  composition  of  motions 
or  forces,  the  resultant  of  any  given  components  is  found  ;  in  the 
resolution  of  motion  or  force,  the  process  is  reversed ;  the  resultant 
being  given,  the  components  are  found,  which  are  equivalent  t  > 
that  resultant. 

If  it  be  required  to  find  what  FIG.  18, 

two  components  can  produce  the 
resultant  A  B  (Fig.  18),  we  have 
only  to  construct  on  A  B,  as  a  base, 
any  triangle  whatever,  as  A  B  C  or 
A  B  D  (Art.  43) ;  then,  if  A  C  is 
one  component,  the  other  is  A  F9 
equal  and  parallel  to  C  B ;  or  if 
A  D  is  one,  the  other  is  A  E,  equal 
and  parallel  to  DB\  and  so  for 

any  triangle  whatever  on  the  base  A  B.     The  number  of  pairs  is 
therefore  infinite,  whose  resultant  in  each  case  is  A  B. 


RESOLUTION    OF    A    FORCE.  33 

• 

The  directions  of  the  components  may  be  chosen  at  pleasure, 
provided  the  sum  of  the  angles  made  with  A  B  is  less  than  two 
right  angles. 

The  magnitude  and  direction  of  one  component  may  be  fixed 
at  pleasure. 

The  magnitudes  of  both  components  may  be  what  we  please, 
provided  their  difference  is  not  greater,  and  their  sum  not  less, 
than  the  given  resultant. 

These  conditions  are  obvious  from  the  properties  of  the 
triangle. 

When  a  given  force  has  been  resolved  into  two  others,  each 
of  those  may  again  be  resolved  into  two,  each  of  those  into  two 
others  still,  and  so  on.  Hence  it  appears  that  a  given  force  may 
be  resolved  into  any  number  of  components  whatever,  with  such 
limitations  as  to  direction  and  magnitude  as  accord  with  the  fore- 
going statements. 

1.  A  motion  of  153  toward  the  north  is  produced  by  the  forces 
100  and  125  ;  how  are  they  inclined  to  the  meridian  ? 

Ans.  54°  28'  and  40°  37'  7". 

2.  A  resultant  of  617  divides  the  angle  between  its  components 
into  28°  and  74°  ;  what  are  the  components  ? 

Ans.   606.34  and  296.14. 

51.  Resolution  of  a  Force,  to  Find  its  Efficiency  in 
a  Given  Direction. — By  the  resolution  of  a  force  into  two 
others  acting  at  right  angles  with  each  other,  it  is  ascertained 
how  much  efficiency  it  exerts  to  produce  motion  in  any  given 
direction.  For  example,  a  weight  W  (Fig.  19),  lying  on  a  hori- 


zontal plane,  and  pulled  by  the  oblique  force  C  A,  is  prevented 
by  gravity  from  moving  in  the  line  0  A,  and  is  compelled  to 
remain  on  the  plane.  Eesolve  C  A  into  C  B,  in  the  plane,  and 
CD  perpendicular  to  it:  then  the  former  represents  the  compo- 
nent which  is  efficient  to  cause  motion  along  the  plane  ;  the  latter 
has  no  influence  to  aid  or  hinder  that  motion  ;  it  simply  dimin- 
ishes pressure  upon  the  plane.  In  like  manner,  if  A  C  is  an 
oblique  force,  pushing  the  weight,  its  horizontal  component,  B  C, 
3 


34 


MECHANICS. 


is  alone  efficient  to  move  it ;  tne  other,  A  B,  merely  increasing 
the  pressure.  In  either  case,  the  whole  force  is  to  that  compo- 
nent which  is  efficient  to  move  the  body  along  the  plane, as  radius 
to  the  cosine  of  inclination.  Also,  the  whole  force  is  to  that  com- 
ponent which  increases  or  diminishes  pressure  on  the  plane,  as 
radius  to  the  sine  of  inclination. 

If  only  88  per  cent,  of  the  strength  of  a  horse  is  efficient  in 
moving  a  boat  along  a  canal,  what  angle  does  the  rope  make  with 
the  line  of  the  tow-path  ?  Ans.  28°  21'  27". 

52.  Resultant  found  by  means  of  Rectangular  Axes.— 

When  several  forces  act  in  one  plane  upon  a  body,  their  resultant 
may  be  conveniently  found  by  the  use  of  right-angled  triangles 
alone.  Select  at  pleasure  two  lines  at  right  angles  to  each  other, 
both  of  them  lying  in  the  plane  of  the  forces,  and  passing  through 
the  point  at  which  the  forces  are  applied.  These  lines  are  called 
axes.  The  following  example  illustrates  their  use  : 

Let  P  A,  P  B,  P  C,  P  D,  P  E  (Fig.  20)  represent  the  forces 
in  Question  5  (Art.  48).  Let  one  axis,  for 
convenience,  be  chosen  in  the  direction  P  A, 
and  let  P  H  be  drawn  at  right  angles  to  it 
for  the  other  axis.  These  axes  are  supposed 
to  be  of  indefinite  length.  Then  proceed  as 
in  Art.  51  to  resolve  each  force  into  two 
components  on  these  axes.  As  P  A  acts  in 
the  direction  of  one  axis,  it  does  not  need  to 
be  resolved.  To  resolve  P  B,  say 


R  :  cos  20C 
R  :  sin  20C 


P  B  :  P  b,  and 
PB:PV: 


again, 


R  :  cos  39°  : :  P  C :  P  c,  and 
R  :  sin  39°  : :  P  C  :  P  c' ,  &c. 

Suppose  P  A  prod  uced  so  as  to  equal  P  A  -f-  Pb  -\-  P  c  -f 
P  d  +  P  e  =  M,  and  P  H  produced  so  as  to  equal  P  b'  +  P  c'  + 
p  d'  +  P  e'  =  N.  Now,  as  M  acts  in  the  line  P  A,  and  N  at 
right  angles  to  it,  their  resultant  and  the  angle  which  it  makes 
with  P  A  are  found  by  the  solution  of  another  right-angled 
triangle.  The  resultant  is  5.1957,  and  the  angle  is  46°  33'  10",  as 
in  Art.  48. 

If  any  components  of  the  resolved  forces  are  opposite  to  P  A 
or  P  H,  they  are  reckoned  as  negative  quantities. 

53.  Analytical  Expression  for  the  Resultant. — Put  A  C 
(Fig.  21)  =  P,  A  B  =  P',  A  D  =  R,  angle  CAB  =  a;  then  in 

triangle  A  B  D  we  have,  by  Geometry,  XZ)2  =  A  B*  +  B  1?  + 


PRINCIPLE    OF    MOMENTS. 


35 


A  B  x  BE,  but  B  E  —  B  D  cos  a  =  P  cos  a,  and  hence  sub- 


stituting as  above  R*  =  P'2  -f-  P2  -f 
2  P'  P  cos  a-,  whence 

It=VP'2  -t-  P2  +  2  PrPc5so' .  (1.) 

Hence,    27*e   resultant   of  any  two 

forces,  acting  at  the  same  point,  is 

equal  to  the  square  root  of  the  sum 

of  the  squares  of  the  two  forces,  plus  twice  the  product  of  the  forces 

into  the  cosine  of  the  included  angle. 

If  a  =  0,  its  cosine  will  be  1,  and  (1)  becomes 

R  =  P  +  P'. 

If  a  =  90°,  its  cosine  will  be  0,  and  we  shall  have 


R  =  VP*  4-  P\ 

If  a  =  180°,  its  cosine  will  be  —  1,  and  we  shall  have 
R  =  P  —  P'. 

1.  Two  forces,  P  and  P  ',  are  equal  in  intensity  to  24  and  30, 
respectively,  and  the  angle  between  them  is  105°  ;  what  is  the 
intensity  of  their  resultant  ?  Ans.  33.21. 

2.  Two  forces,  P  and  P',  whose  intensities  are,  respectively, 
equal  to  5  and  12,  have  a  resultant  whose  intensity  is  13  ;  required 
the  angle  between  them.  An.  90°. 

3.  A  boat  is  impelled  by  the  current  at  the  rate  of  4  miles  per 
hour,  and  by  the  wind  at  the  rate  of  7  miles  per  hour  ;  what  will 
be  her  rate  per  hour  when  the  direction  of  the  wind  makes  an 
angle  of  45°  with  that  of  the  current  ?  Ans.  10.2  miles. 

4.  Two  forces  and  their  resultant  are  all  equal  ;  what  is  the 
value  of  the  angle  between  the  two  forces  ?  Ans.  120°. 


FIG.  22. 


54.  Principle  of  Moments. — The  moment  of  a  force,  with 
respect  to  a  point,  is  the  product 
of  the  force  into  the  perpendicu- 
lar let  fall  from  the  point  to  the 
line  of  direction  of  the  force. 

The  fixed  point  is  called  the 
centre  of  moments ;  the  perpen- 
dicular distance,  the  lever-arm 
of  the  force  ;  and  the  moment 
measures  the  tendency  of  the 
force  to  produce  rotation  about 
the  centre  of  moments. 

Denote  the  forces  (Fig.  22) 
by  P,  P'  and  their  resultant  by 
R.  From  E  any  point  in  the 


36 


MECHANICS. 


plane  of  the  forces  let  fall  upon  the  directions  of  the  forces 
the  perpendiculars  E a,  EC,  Ed.  Represent  these  by  Z,  L,  I'. 
Draw  two  rectangular  axes  of  reference  as  in  Art.  52,  so  that 
one  of  them  may  pass  through  A  and  E.  The  projection  of  the 
resultant  R  is  equal  to  the  sum  of  the  projections  of  its  compo- 
nents (Art.  52) ;  hence, 

A  Y  =  A  Y"  +  A  Y'     .'"'.     ;     .     .     (1) 

By  similar  triangles  Ac  E  and  A  R  Y,  we  have 

LxR. 

j^   I   —   J~ICT~  > 


A  Y:  EC=L  ::  A  R  =  R  :  A  E    .: 
by  similar  triangles  A  a  E  and  A  P  Y",  we  have 

A  Y"  :  Ea  =  I  ::  A  P  =  P  :  A  E    .-.  A  Y"  = 
and  by  similar  triangles  Ad  E  and  A  P'  Y',  we  have 

AY':Ecl  =  l'::AP'  =  P':AE     .'.  A  Y'  = 


AE 

I  x  P 
AE   ' 

r  x  P' 


FIG.  23. 


substituting  these  values  in  Eq.  (1)  and  multiplying  by  A  E,  we 
have 

R  x  L  =  P  x  I  +  P'  x  I'. 

Hence,  the  moment  of  the  resultant  of  two  forces  with  respect  to 
any  point  is  equal  to  the  algebraic  sum  of  the  moments  of  the  forces 
taken  separately. 

By  using  the  resultant  as  above  and  a  third  force  the  moment 
of  the  resultant  of  the  three  forces  may  be  proved  equal  to  the 
algebraic  sum  of  the  moments  of  the  forces,  and  so  on  for  any 
number  of  forces. 

To  illustrate  the  application  of  the  principle  of  moments,  sup- 
pose a  weight  W  of  1000  Ibs.  to  be  suspended  from  the  end  of  a 
spar  D  E  as  in  the  figure  ; 
required  the  strain  upon  the 
stay  E  B. 

We  have  three  forces  in 
equilibrium  acting  at  E,  viz., 
the  weight  W,  the  strain  upon 
the  rope  8,  and  the  upward 
thrust  of  the  spar  T.  If  we 
select  the  centre  of  moments 
upon  the  line  of  either  force, 
the  moment  of  that  force  will 
be  zero.  As  the  thrust  J'is  equal  and  opposed  to  the  resultant 
of  8  and  W,  we  will  take  D  as  the  centre  of  moments,  and  we 
have, 

Moment  of  T  =  moment  of  S  +  moment  of  W ;  but  T  x  0  = 
moment  of  T,  S  x  KD  =  moment  of  S,  and  W  x  D  n  =  mo- 


PARALLEL    FORCES.  37 

ment  of  W.  But  W  tends  to  cause  E  to  revolve  towards  the  right 
about  the  point  D9  while  8  tends  to  cause  revolution  of  E  towards 
the  left ;  hence  one  must  be  regarded  as  a  positive  and  the  other 
as  a  negative  moment,  and  we  have,  finally,  if  we  call  W  x  D  n 

positive,  0  =  —  8  x  KD  +  W  x  n  D,  whence,  S  =  — -£-^ — 

If  in  the  problem,  D  E  =  20  ft,  B  D  =  20  ft,  and  E  B  D  =  30°, 
then  will  KD  —  BD  sine  30°  =  10,  and  D  n  =  D  e  sine  30°  =  10. 

_%jS=  1000.X20  =  looolbs.      g||| 

To  find  the  thrust  at  D,   take  the  point  B  as  a  centre  of 
moments,  then 

T  x  Ba  =  S  x  0  +  W  x  Bn  ;  /.  T=  W*  Bn  • 

nd 

Now  B  n  =  B  D  +  D  n  =  30 ;   to  find  B  a,  we  have 


K  E  =      DE'—KD*  =  <v/300,  and  B  E  =  2  K  E  =  2A/300. 
In  similar  triangles  K  D  ^and  B  E  a  we  have 

D  E\  B  E  ::  KD  :  Ba\ 
2V300~  X  10  1000  x  30 

•'•  B a  =  ~~20~  =  'V/300  5  hence  T  =  V  300 
Remember  that  when  three  forces,  acting  at  the  same  point, 
are  in  equilibrium,  one  of  the  three  being  known,  either  of  the 
other  two  can  be  found  by  taking  the  centre  of  moments  on  the 
line  of  the  force  not  sought,  and  equating  the  moments  of  the  two 
forces  considered. 

55.  Forces  Acting  at  Different  Points.  Parallel 
Forces. — We  have  thus  far  considered  forces  acting  upon  a 
single  particle,  or  upon  one  F  . 

point  of  a  body.  If,  how- 
ever, two  forces  P  and  P', 
in  the  same  plane,  act  upon 
A  and  B,  two  different 
points  of  a  rigid  body,  they 
may  still  have  a  resultant. 

Let  the  lines  of  direc- 
tions of  the  two  forces  A  F  and  B  D  (Fig.  24)  be  produced  to 
meet  in  C.  The  two  forces  may  then  be  considered  as  acting  at 
(7,  and  thus  compounded  into  a  single  force  at  that  point,  or  at 
the  point  G  of  the  body. 

Calling  the  angle  B  C  G=$  and  A  CG=awe  have,  projecting 
P'  and  P  upon  the  line  of  R, 

R  =  P1  cos  0  +  P  cos  a  .  .  .  (1). 


38 


MECHANICS. 


When  the  forces  become  parallel,  as  A  F  and  B  E,  (3  =  0,  and 
o  =  0,  and  (1)  becomes 

R  =  P'  +  P  .  . .  (2). 

If  the  parallel  forces  act  in  opposite  directions,  as  A  F  and 
B  E',  then  a  =  180°,  and  ft  =  0,  and  (1)  becomes 

R  =  P'  -  P .  .  .  (3).     Hence, 

TJie  resultant  of  two  parallel  forces  is  in  a  direction  parallel  to 
them  and  equal  to  their  algebraic  sum. 

56.  Point  of  Application  of  the  Resultant. — Let  P  and 
P'  (Figs.  25,  26)  be  two  parallel  forces  acting  in  the  same  or  in 

FIG.  25.  FIG.  26. 


opposite  directions,  and  let  E  be  the  point  of  application  of  the 
resultant.  Assume  this  point  as  a  centre  of  moments  ;  then  from 
Art.  54,  since  L  =  0, 

P  x  H  E  =  P'  x  G  E,  or,  in  the  form  of  a  proportion, 
P'  :  P  : :  H  E  :  G  E.     But  by  similar  triangles, 
HE:  GE  ::AE  :E  B',.: 
P'  :  P  : :  A  E  :  E  B. 

That  is,  the  line  of  direction  of  the  resultant  of  two  parallel  forces 
divides  the  line  joining  the  points  of  application  of  the  components, 
inversely  as  the  components. 

By  composition  (Fig.  25)  and  division  (Fig.  26)  we  obtain 
P'  +  P  :  P  : :  A  B  :  E  B,  and 
P  -  P'  :P::A  B  :  E  B. 

That  is,  if  a  straight  line  be  drawn  to  meet  the  lines  of  two  parallel 
forces  and  their  resultant,  each  of  the  three  forces  will  be  propor- 
tional to  that  part  of  the  line  contained  between  the  other  two. 
When  the  forces  act  in  the  same  direction,  we  have 

P  x  A  B 
E  B  =    ^,   .  '  ,„ ,  and  when  they  act  in  opposite  directions, 


E  B  = 


P'+  p> 
P  x  AB 


P-P'  . 

If,  in  the  last  case,  P  —  P',  then  E  B  will  be  infinite.  The 
two  forces  in  this  case  constitute  what  is  called  a  couple.  Their 
effect  is  to  produce  rotation  about  a  point  between  them. 

Any  number  of  parallel  forces  may  be  reduced  to  a  single  force 


EQUILIBRIUM   OF  FORCES. 


39 


(or  to  a  couple)  by  first  finding  the  resultant  of  two  forces,  then 
the  resultant  of  that  and  a  third  force,  and  so  on  to  the  last.  And 
any  single  force  may  be  resolved  into  two  or  any  number  of  paral- 
lel forces  by  a  method  the  reverse  of  this. 

57.  Equilibrium  of  Parallel  Forces. — In  order  that  a  force 
may  be  in  equilibrium  with  two  parallel  forces, 

1.  It  must  be  parallel  to  them. 

2.  It  must  be  equal  to  their  algebraic  sum. 

3.  The  distances  of  its  line  of  action  from  the  lines  in  which  the 
two  forces  act,  must  be  inversely  as  the  forces. 

These  three  conditions  belong  to  the  resultant  of  two  parallel 
forces,  and  therefore  belong  to  that  force  which  is  in  equilibrium 
with  the  resultant. 

58.  Equilibrium  of  Couples. — If  two  parallel  forces  are 
such  as  to  constitute  a  couple,  no  one  force  can  be  in  equilibrium 
with  them.     For  the  resultant  of  a  couple  is  zero,  and  has  its  point 
of  application  at  an  infinite  distance  (Art.  56).     But  a  couple  can 
be  held  in  equilibrium  by  another  couple  ;  and  the  second  couple 
may  be  either  larger  or  smaller  than  the  given  couple,  or  it  may 
be  equal  to  it. 

Let  the  couple  P  and  P'  (Fig.  27)  act  FIG.  27. 

on  a  body  at  the  points  A  and  B ;  they 
tend  to  produce  rotation  about  the  middle 
point  C.  If  another  couple,  Q  and  Q', 
equal  to  P  and  P',  should  be  applied  to 
produce  equilibrium,  one  must  directly 
oppose  P,  and  the  other  P'.  Then  A  and 
B,  being  each  held  at  rest,  all  the  forces 
are  in  equilibrium. 

But  if  the  second  couple  is  less  than 
P  and  P',  they  must  act  at  distances  from 
(7,  which  are  as  much  greater  as  the  forces 
are  less  ;  or,  if  the  second  couple  is  greater 
than  the  first,  they  must  act  at  distances  — H1-E7 

which  are  as  much  less.  Thus,  the  couple 
p  and  p,  acting  at  D  and  E,  tend  to  produce  rotation  about  C  in 
one  direction,  and  P  and  P'  in  the  opposite ;  and  these  tendencies 
are  equal  when  D  G  :  A  C : :  P  :  p.  For,  since  the  opposite  forces, 
P  and  p,  are  inversely  as  their  distances  from  (7,  their  resultant  is 
at  (7,  and  is  equal  to  P  —  p  (Art.  55).  For  the  same  reason,  the 
resultant  of  P'  and  p'  is  at  C,  and  equal  to  P'  —  p'.  But  P— p  = 
P'  —  p',  and  they  act  in  opposite  directions.  Hence  C  is  at  rest, 
and  therefore  all  the  forces  are  in  equilibrium. 

59.  The  Parallelepiped  of  Forces.— Hitherto  forces  have 


p-p 


CP-p' 


40 


MECHANICS. 


FIG.  28. 


been  considered  as  acting  in  the  same  plane.  But  if  forces  act  in 
different  planes,  the  solution  of  every  case  may  be  reduced  to  the 
following  principle,  called  the  parallelepiped  of  forces. 

Any  three  forces  acting  in  different  planes  upon  a  body  may  be 
represented  ly  the  adjacent  edges  of  a  parallelepiped,  and  their  re- 
sultant ~by  the  diagonal  which  passes  through  the  intersection  of 
those  edges. 

Let  A  C,  A  D,  and  A  E  (Fig.  28),  be  three  forces  applied  in 
different  planes  to  the  body  at  A. 
Construct  the  parallelepiped  G  P, 
having  A  C,  A  D,  and  A  E>  for  its 
adjacent  edges,  and  from  A  draw  the 
diagonal  A  B.  The  section  through 
the  opposite  edges  A  C  and  P  B  is  a 
parallelogram,  and  therefore  A  B  is 
the  resultant  of  A  C  and  A  P,  and 
A  P  is  the  resultant  of  A  D  and  A  E. 
Hence  A  B  is  the  resultant  of  A  C, 
E. 


This  process  may  obviously  be  reversed,  and  a  given  force  may 
be  resolved  into  three  components  in  different  planes  along  the 
edges  of  a  parallelepiped,  having  such  inclinations  as  we  please. 

60.  Rectangular  Axes.  —  The  parallelepiped  generally 
chosen  is  that  whose  sides  are  rectangles  ;  the  three  adjacent 
edges  of  such  a  solid  are  called  rectangular  axes.  All  the  forces 
which  can  possibly  act  on  a  body  may  be  resolved  into  equivalent 
forces  in  the  direction  of  three  such  axes.  And  since  all  forces 
which  act  in  the  direction  of  any  one  line  may  be  reduced  to  a 
single  force  by  taking  their  algebraic  sum,  therefore  any  number 
of  forces  acting  through  one  point  may  be  reduced  to  three  in  the 
direction  of  three  axes  chosen  at  pleasure. 

FIG.  29.  FIG.  30. 


Let  A  X,  A  Y  (Fig.  29)  be  at  right  angles  with  each  other, 
and  A  Z  perpendicular  to  the  plane  A  X  and  A  Y.     Let  A  B 


COMPONENTS  AND  RESULTANT. 


41 


represent  a  force  acting  on  A.  Eesolve  A  B  into  A  C  on  the  axis 
A  Z,  and  A  P  in  the  plane  of  A  X,  A  Y ;  then  resolve  A  P  into 
A  D  and  A  E  on  the  other  two  axes.  Therefore,  A  C,  A  D,  and 
A  E  are  three  rectangular  forces,  whose  resultant  is  A  B. 

Let  the  axes  A  X,  AY,AZ,  be  produced  indefinitely  (Fig.  30) 
to  X',  Y',  Z' ;  then  their  planes  will  divide  the  angular  space 
about  A  into  eight  solid  right  angles,  namely:  A-X  Y  Z,  A-X  Y'  Z, 
A-X'Y'Z,  A-X'YZ,  above  the  plane  of  Xand  Y9  and  A-XYZ', 
A-XY1  Z',  A-X'  Y'  Z',  A-X'  Y  Z'  below  it. 


FIG.  81. 


61.    Geometrical   Relation   of  Components  and  Re- 

sultant. —  A  force  acting  on  the  body 
A  may  be  situated  in  any  one  of  the 
eight  angles,  and  its  value  may  be 
expressed  in  terms  of  the  squares  of 
its  three  components.  Let  A  B 
(Fig.  31)  be  resolved  as  before  into 
the  rectangular  components  A  C, 
A  D,  and  A  E.  Then,  by  the  right- 
angled  triangles,  we  find 


and 


'JO 


and  A  B  = 


AD*  '  +  A  E\ 


If  A  B  is  in  the  plane  of  X  and  Y,  the  component  on  the  axis 
of  /?  becomes  zero,  and  A  B  =  \/A  C2  -f  A  D*,  and  similarly  for 
the  other  planes. 

62.  Trigonometrical  Relation  of  Components  and  Re- 

sultant. —  Let  the  angles  which  A  B  makes  with  the  axes  of 
X,  Y,  Z,  respectively,  be  a,  j3,  y  ;  that  is,  B  A  0  =  a,  B  A  D  =  )3, 
B  A  E  =  y.  In  the  triangle  ABC,  right-angled  at  (7,  we  have 
A  B  :  A  C  :  :  rad  :  cos  a  ;  therefore,  making  rad  =  1, 

A  C  =  A  B  .  cos  a. 
In  like  manner,      A  D  =  A  B  .  cos  (3  ; 
and  A  E  =  A  B  .  cos  y. 

And  since  A  B  is  the  resultant  of  the  forces  AC,  AD,  and 
A  E,  it  is  the  resultant  of  A  B  .  cos  a,  A  B  .  cos  ft  A  B  .  cos  y. 
In  general,  the  components  of  any  force  P,  when  resolved  upon 
three  rectangular  axes,  are  P  .  cos  a,  P  .  cos  0,  P  .  cos  y. 


42  MECHANICS. 

63.  Any  Number  of  Forces  Reduced  to  Three  on 
Three  Rectangular  Axes. — Suppose  the  body  at  A  to  be  acted 
upon  by  a  second  force  P'9  whose  direction  makes  with  the  axes 
the  angles  a',  p,  y' ;  then,  as  before,  P'  is  the  resultant  of  P' .  cos  a', 
P .  cos  /3',  P' .  cos  y' ;  and  a  third  force  P'',  in  like  manner,  has 
for  its  components  P"  .  cos  a",  P"  .  cos  j3",  P"  .  cos  y"  ;  and  so  of 
any  number  of  forces. 

Now,  all  the  components  on  one  axis  may  be  reduced  to  one 
force  by  adding  them  together.  Hence,  the  whole  force  in  the 
axis  of  X  =  P .  cos  a  +  P' .  cos  a'  +  P" .  cos  a"  -j-  P'" .  cos  a'"  +  &c. ; 
the  whole  in  the  axis  of  Y, 

=  P .  cos  j3  +  P'  -  cos  j3f  +  P" .  cos  0"  +  P'" .  cos  0" '  +  &c. ; 
and  that  in  the  axis  of  Z9 

=  P .  cos  y  +  P' .  cos  y'  +  P" .  cos  y"  -f  P'" .  cos  y'"  +  &c. 

If  any  component  acts  in  a  direction  opposite  to  others  in  the 
same  axis,  it  is  aifected  by  a  contrary  sign,  so  that  the  force  in  the 
direction  of  any  axis  is  the  algebraic  sum  of  all  the  individual 
forces  in  that  axis. 

If  the  sum  of  the  components  in  one  axis  is  reduced  to  zero  by 
contrary  signs,  the  effect  of  all  the  forces  is  limited  to  the  plane 
of  the  other  axes,  and  is  to  be  obtained  as  in  Art.  52,  where  two 
axes  were  employed.  If  the  sum  of  the  components  on  each  of 
two  axes  is  reduced  to  zero,  then  the  whole  force  is  exerted  in  the 
direction  of  the  remaining  axis,  and  is  therefore  perpendicular  to 
the  plane  of  the  other  two. 

64.  Equilibrium  of  Forces  in  Different  Planes. — Since 
all  the  forces  which  can  operate  on  a  body  may  be  reduced  to 
three  forces  on  rectangular  axes,  it  is  obvious  that  the  whole  sys- 
tem of  forces  cannot  be  in  equilibrium  till  the  sum  of  the  compo- 
nents on  each  axis  is  reduced  to  zero.  We  must  have,  therefore, 
in  Art.  63,  as  conditions  of  equilibrium,  these  three  equations  for 
the  three  axes,  X,  Y,  Z : 

P .  cos  a  +  P' .  cos  a'  +  P"  .  cos  a"  +,  Ac.,  =  0  ; 
P  .  cos  0  +  P' .  cos  0'  +  P"  .  cos  0"  +  ,  &c.,  =  0  ; 
P .  cos  y  +  P' .  cos  y'  +  P"  .  cos  y"  +  ,  &c.,  =  0. 

65.  Forces  Resisted  by  a  Smooth  Surface.— When- 
ever any  forces  cause  pressure  upon  a  surface  without  friction, 
and  are  held  in  equilibrium  by  its  resistance,  the  resultant  of 
those  forces  must  be  at  right  angles  to  the  surface.  Suppose  that 


CENTRE   OF    GRAVITY. 


43 


AIG.  32. 


D  A  (Fig.  32)  is  either  a  single  force  or  the  resultant  of  two  or 
more  forces,  and  that  it  is  held  in  equili- 
brium by  the  reaction  oiAB,  a  smooth  sur- 
face. If  D  A  is  not  perpendicular  to  the 
surface,  it  can  be  resolved  into  two  com- 
ponents, one  perpendicular  to  the  surface 
A  B,  the  other  parallel  to  it.  The  former, 
C  A,  is  neutralized  by  the  resistance  of 
the  surface ;  the  latter,  B  A,  is  not  re- 
sisted, and  produces  motion  parallel  to 
the  surface,  contrary  to  the  supposition.  Therefore  D  A,  if  held 
in  equilibrium  by  the  surface  A  B,  must  be  perpendicular  to  it. 


CHAPTER    IV. 

THE  CENTRE  OF  GRAVITY. 

66.  The  Centre  of  Gravity  Defined. — In  every  body  and 
in  every  system  of  bodies,  there  is  a  point  so  situated  that  all  the 
parts  acted  on  by  the  force  of  gravity  balance  each  other  about  it 
in  every  position.    That  point  is  called  the  centre  of  gravity.    The 
force  of  gravity  acts  in  parallel  lines  on  every  particle  of  a  body ; 
the  centre  of  gravity  must  therefore  be  the  point  through  which 
the  resultant  of  all  these  parallel  forces  is  directed,  in  every  posi- 
tion of  the  body.     Hence,  if  the  centre  of  gravity  is  supported, 
the  body  is  supported.     As  to  the  support  of  the  body,  therefore, 
we  may  imagine  all  parts  of  it  to  be  collected  in  its  centre  of 
gravity.     When  a  system  of  bodies  is  considered,  they  are  con- 
ceived to  be  united  to  each  other  by  inflexible  rods,  which  are 
without  weight. 

67.  Centre  of  Gravity  of  Equal  Bodies  in  a  Straight 
Line. — The  centre  of  gravity  of  two  equal  particles  is  in  the 
middle  point  between  them.     Let  A  and  B  (Fig.  33),  two  equal 
particles,  be  joined  by  a  straight  line,  and  let 

A  a  and  B  b  represent  the  forces  of  gravity. 
The  resultant  of  these  forces,  since  they  are 
parallel  and  equal,  will  pass  through  the  mid- 
dle of  A  B  (Art.  56) ;  G  is  therefore  the  centre 
of  gravity.  In  like  manner  it  is  proved  that 
the  centre  of  gravity  of  two  equal  bodies  is 
in  the  middle  point  between  their  respective  centres  of  gravity. 


FIG.  33. 


p--^G 

I  **""'•— i] 


44 


MECHANICS. 


Any  number  of  equal  particles  or  bodies,  arranged  at  equal 
distances  on  a  straight  line,  have  their  common  centre  of  gravity 
in  the  middle ;  since  the  above  reasoning  applies  to  each  pair, 
taken  at  equal  distances  from  the  extremes.  Hence,  the  centre 
of  gravity  of  a  material  straight  line  (e.g.,  a  fine  straight  wire)  is 
in  the  middle  point  of  its  length. 

68.  Centre  of  Gravity  of  Regular  Figures. — In  the  dis- 
cussion of  the  centre  of  gravity  in  relation  to  form,  bodies  are  con- 
sidered uniformly  dense,  and  surfaces  are  regarded  as  thin  laminae 
of  matter. 

In  plane  figures  the  centre  of  gravity  coincides  with  the  centre 
of  magnitude,  when  they  have  such  a  degree  of  regularity  that  there 
are  two  diameters,  each  of  which  divides  the  figure  into  equal  and 
symmetrical  parts. 

The  circle,  the  parallelogram,  the  regular  polygon,  and  the 
ellipse,  are  examples. 

For  instance,  the  regular  hexagon  (Fig.  34)  is  divided  sym- 
metrically by  A  B,  and  also  by  C  D.  Conceive 
the  figure  to  be  composed  of  material  lines 
parallel  to  A  B.  Each  of  these  has  its  centre 
of  gravity  in  its  middle  point,  that  is,  in  C  D, 
which  bisects  them  all  (Art.  67).  Hence,  the 
centre  of  gravity  of  the  whole  figure  is  in  C  D. 
For  the  same  reason  it  is  in  A  B.  It  is,  there- 
fore, at  their  intersection,  which  is  also  the 
centre  of  magnitude. 

By  a  similar  course  of  reasoning  it  is  shown  that  in  solids  of 
uniform  density,  which  are  so  far  regular  that  they  can  be  divided 
symmetrically  by  three  different  planes,  the  centres  of  gravity  and 
magnitude  coincide  ;  e.g.,  the  sphere,  the  parallelepiped,  the 
cylinder,  the  regular  prism,  and  the  regular  polyhedron. 

69.  Centre  of  Gravity  between  Two  Unequal  Bodies.— 

The  centre  of  gravity  of  two  unequal  bodies  is  in  a  straight  line 

joining  their  respective  centres  of  gravity,  and  at  the  point  which 

divides  their  distance  in  the  inverse  ratio  of  their  weights.     Lot 

A  a  and  B  b  (Fig.  35),  passing  through  the 

centres  of  gravity  of  A  and  B,  be  proportional 

to  their  weights,  and  therefore  represent  the 

forces  of  gravity  exerted  upon  them.     By  the 

laws  of  parallel  forces,  the  resultant  G  g  = 

A  a  +  B  b  (Art.  55),  and  A  a  :  B  I : :  B  G : 

A  G.    Therefore  the  centre  of  gravity  must 

be  at  G,  through  which  the  resultant  passes 


FIG.  35. 


CENTRE   OF    GRAVITY. 


45 


(Art.   66).      This  obviously  includes  the  case  of  equal  weights 
(Art.  67). 

It  appears  from  the  foregoing  that  the  whole  pressure  on  a 
support  at  6ris  A  -f-#,  and  that  the  system  is  kept  in  equilibrium 
by  such  support. 

70.  Equal  Moments  with  Respect  to  the  Centre  of 

Gravity.  —  Applying  the  principle 
of  moments  we  have,  calling  the 
weights  TFand  W,  and  taking  the 
centre  of  moments  at  G  (Fig.  36) 


FIG.  36. 


but  G  x  :  G  y  : 
.-.  W  x  A  G  =  W 
proved  in  Art.  56. 


A  G  :  G  B  ; 
x  G  B,  as  was 


FIG,  37. 


•  B 


71.  Centre    of    Gravity    between    Three    or    More 
Bodies.  —  The  method  of  determining  the  centre  of  gravity  of 
two  bodies  may  be  extended  to  any  number. 

Let  A,  B,  'C,  D,  &c.  (Fig.  37),  be  the  weights  of  the  bodies, 
and  let  the  centres  of  gravity  of  A  and  B  be 
connected    together  by  the    inflexible    line 
A  B. 

Divide  A  B  so  that  A  :  B  :  :  B  G  :  A  G, 
or  A  +  B  :  B  ::  A  B  :  A  #;  then  G  is  the 
centre  of  gravity  of  A  and  B.  Join  C  G  ; 
and  since  A  -\-  B  may  be  considered  as  at  the  point  G,  divide 
C  G  so  that  A  +  B  +  C  :  0  ::  C  G  :  G  g.  In  like  manner,  K, 
the  centre  of  gravity  of  four  bodies,  is  found  by  the  proportion, 
A  +  B  -f-  C  +  D  :  D  :  :  D  g  :  g  K.  The  same  plan  may  be  pur- 
sued for  any  number  of  bodies. 

72.  Centre  of  Gravity  of  a  Triangle.—  The  centre  of 
gravity  of  a  triangle  is  one-third  of  the  distance  from  the  middle 
of  a  side  to  the  opposite  angle.    Bisect  A  C  in  D  (Fig.  38),  and 
B  C  in  E\  join  A  E,  B  D,  and  D  E.     B  D 

bisects  all  lines  across  the  triangle  parallel  to 
A  G\  therefore  the  centre  of  gravity  of  all  those 
lines  —  that  is,  of  the  triangle—  is  in  B  D.  For 
a  like  reason,  it  is  in  A  JEJ,  and  therefore  at  their 
intersection,  G.  Since  E  C  —  £  B  C,  and  D  C 
=  $  A  C,  /.  E  D  =  |  A  B.  BntEGD  and 
A  G  B  are  similar  ;  .-.  D  G  :  B  G  :  :  D  E  : 
AB::l:2;  .:  DG  —  ^BG 


Fia  38. 


73.  Centre  of  Gravity  of  an  Irregular  Polygon.  —  Divide 
the  polygon  into  triangles  by  diagonals  drawn  through  one  of  its 


46 


MECHANICS. 


FIG.  39. 


angles,  and  then  proceed  according  to  the  methods  already  given. 
Let  A  C E  (Fig.  39)  be  an  irregular  polygon,  whose  centre  of  grav- 
ity is  to  be  found.  Divide 
it  into  the  triangles  P,  Q, 
R,  S,  by  diagonals  through 
A,  and  find  their  centres  of 
gravity  a,  b,  c,  d  (Art.  72). 
Join  a  b,  and  divide  it  so 
that  a  b  :  a  G  : :  P  +  Q  :  Q; 
then  G  is  the  centre  of 
gravity  of  the  quadrilateral 
P  -f  Q.  Then  join  G  c,  and 


c:  Gg  \\ 

:  R.     By  proceeding  in  this 
manner  till  all  the  triangles 
are  used,  the  centre  of  gravity  of  the  polygon  is  found  at  the  last 
point  of  division. 

74.  Centre  of  Gravity  of  the  Perimeter  of  an  Irregu- 
lar Polygon. — Find  the  centre  of  gravity  of  each  side,  which  is 
at  its  middle  point,  and  then 

proceed  as  in  Art.  71,  the 

weight  of  each  line  being 

considered  proportional  to 

its  length.     Thus,  let  a,  b, 

c,   &c.,   be  the  centres  of 

gravity  of  the  sides,  A  B, 

B  C,  CD,  &c.  (Fig.  40);  B 

join  a  b,  and  divide  it  so 

that  ab:aG::AB  +  BG 

:  B  G ;  then  G  is  the  centre 

of  gravity  of  A  B  and  B  C. 

Next  join  G  c,  and  make  Gc:  Gg  : :  AB  +  BC  +  CD :  CD; 

then  g  is  the  centre  of  gravity  of  those  three  sides.     Proceed  in 

this  manner  till  all  the  sides  are  used. 

The  perimeter  of  a  polygon  having  the  degree  of  regularity  de- 
scribed in  Art.  68,  has  its  centre  of  gravity  at  the  centre  of  the 
figure,  as  may  be  easily  proved.  If  a  polygon  has  a  less  degree  of 
regularity  than  that,  the  centre  of  gravity  both  of  its  area  and  its 
perimeter  may  usually  be  found  by  methods  more  direct  and 
simple  than  those  given  for  polygons  wholly  irregular. 

75.  Centre  of  Gravity  of  a   Pyramid. — The  centre  of 
gravity  of  a  triangular  pyramid  is  in  the  line  joining  the  vertex 
and  the  centre  of  gravity  of  the  base,  at  one-fourth  of  the  distance 
from  the  base  to  the  vertex. 


CENTRE    OF    GRAVITY. 


47 


Let  G  (Fig.  41)  be  the  centre  of  gravity  of  the  base  B  D  G ;  and 
g  that  of  the  face  ABC.  The  line  A  G  passes  through  the  centre 
of  gravity  of  every  lamina  FIG  41 

parallel  to  D  B  C,  on  account 
of  the  similarity  and  similar 
position  of  all  those  laminae ; 
/.  the  centre  of  gravity  of  the 
pyramid  is  in  A  G.  For  a 
similar  reason,  it  is  in  Dg; 
and  therefore  at  their  inter- 
section, 0.  ^owEG=%ED, 
and  Eg—\EA\  hence,  by  D 
similar  triangles,  g  G=^AD. 
But  Gg  0  and  A  0  D  are  also 
similar ;  .-.  G  0  =  |  A  0  = 

From  this  it  is  readily  proved  that  the  centre  of  gravity  of 
every  pyramid  and  cone  is  one-fourth  of  the  distance  from  the 
centre  of  gravity  of  the  base  to  the  vertex. 

76.  Examples  on  the  Centre  of  Gravity. — 

1.  A,  B,  and  C  (Fig.  42),  weigh,  respectively,  3,  2,  and  1 

pounds,  A  B  =  5  ft,  B  C  =  4  ft.,  and 

G  A  =  2  ft.      Find    the   distance   of 

their  centre  of  gravity  from  C. 

First,  from  the  given  sides  of  the 

triangle  ABC,  calculate  the  angles. 

A  is  found  to  be  49°  27|'.     Next  find 

the  place  of  G,  the  centre  of  gravity  of 

A  and  B,  by  the  proportion,  A  +  B  : 


B  : :  A  B  :  A  G  :  A  G  is 


2  ft.,  equal  to  A  C.  Calculate  C  G,  the  base  of  the  isosceles  tri- 
angle AGO.  Its  length  is  1.G73.  Then  find  G  g  by  the  propor- 
tion C'GiCgnA+B  +  CiA  +  B;  therefore  Cg  =  1.394. 

2.  A  =  5  Ibs.,  B  —  3  Ibs.,   and   C  =  12  Ibs.;    A  B  =  S  ft., 
A  0  =  4  ft.,  and  the  angle  A  is  90° ;  find  the  distance  of  the 
centre  of  gravity  of  A,  B,  and  C  from  C.  Ans.  2  ft. 

3.  Three  equal  bodies  are  placed  at  the  angles  of  any  triangle 
whatever  ;  show  that  the  common  centre  of  gravity  of  those  bodies 
coincides  with  the  centre  of  gravity  of  the  triangle. 

4.  Find  the  centre  of  gravity  of  five  equal  heavy  particles 
placed  at  five  of  the  angular  points  of  a  regular  hexagon. 

Ans.  It  is  one-fifth  of  the  distance  from  the  centre  to 
the  third  particle. 

5.  A  regular  hexagon  is  bisected  by  a  line  joining  two  oppo- 
site angles  ;  where  is  the  centre  of  gravity  of  one-half  ? 

Ans.  Four-ninths  of  the  distance  from  the  centre  to  the 
middle  of  the  second  side. 


48  MECHANICS. 

6.  A  square  is  divided  by  its  diagonals  into  four  equal  parts, 
one  of  which  is  removed  ;  find  the  distance  from  the  opposite 
side  of  the  square  to  the  centre  of  gravity  of  the  remaining  figure. 

Am.  T\  of  the  side  of  the  square. 

7.  Two  isosceles  triangles  are  constructed  en  opposite  sides  of 
the  same  base,  the  altitude  of  the  greater  being  h,  and  of  the  less, 
/*' ;  where  is  the  centre  of  gravity  of  the  whole  figure  ? 

Ans.  On  the  altitude  of  the  greater  triangle,  at  a  distance 
from  the  common  base  equal  to  -J-  (Ji  — -  h'). 

8.  The  base  and  the  place  of  the  centre  of  gravity  of  a  triangle 
being  given,  required  to  construct  the  triangle. 

9.  Given  the  base  and  altitude  of  a  triangle  ;  required  to  con- 
struct the  triangle,  when  its  centre  of  gravity  is  perpendicularly 
over  one  end  of  the  base. 

10.  On  a  cubical  block  stands  a  square  pyramid,  whose  base, 
volume,  and  mass  are  respectively  equal  to  those  of  the  cube; 
where  is  the  centre  of  gravity  of  the  figure  ? 

Ans.  One-eighth  of  the  height  of  the  c«be  above  its 
upper  surface. 

77.  Centre  of  Gravity  of  Bodies  in  a  Straight  Line 
referred  to  a  Point  in  that  Line. — If  several  bodies  are  in  a 
straight  line,  their  common  centre  of  gravity  may  be  referred  to  a 
point  in  that  line  ;  and  its  distance  from  that  point  is  obtained  by 
multiplying  each  weight  into  its  own  distance  from  the  same  goint, 
and  dividing  the  sum  of  the  products  by  the  sum  of  the  weights. 
Let  A,  By  (7,  and  D,  represent  the  weights  of  several  bodies,  whose 
centres  of  gravity  are  in  the  straight  line  o  D  (Fig.  43).   Required 

FIG.  43. 
o  A  B  c  D 

i « •— ^ . 9 

G 

the  distance  of  their  common  centre  of  gravity  from  any  point  o 
assumed  in  the  same  line.  Let  G  be  their  common  centre  of 
gravity  ;  then,  calling  R  the  resultant  of  the  several  weights  A,  B, 
C  and  D,  which  acts  at  the  point  G,  we  have  from  principle  of 
moments, 

R  xoG  =  A  xAo  +  BxBo+Cx  C o  -f  D  x  D  o, 
and  since        R  =  A  4-  B  +  C  +  D  (Art  55),   we  have 

AxAo  +  BxBo+CxCo  +  DxDo 
A +£+  C+D 

78.  Centre   of  Gravity  of  a   System   referred   to  a 
Plane. — If  the  bodies  are  not  in  a  straight  line,  they  may  be  re- 
ferred to  a  plane,  which  is  assumed  at  pleasure.     The  distance  of 


CENTRE    OF    GRAVITY. 


FIG.  44. 


S/ 


their  common  centre  of  gravity  from  that  plane  is  expressed  as 
before  :  multiply  each  weight  into  its  own  distance  from  the  plane, 
and  divide  the  sum  of  the  products  by  the  sum  of  the  bodies. 

Let  p,  p'  p"  (Fig.  44),  represent  the  weights  of  several  bodies, 
whose  centres  of  gravity  are  at  those  points  respectively,  and  let 
A  O  be  the  plane  of  reference. 
Join  p  p',  and  let  g  be  the  com- 
mon centre  of  gravity  of  p  and 
p''9  draw  p  x,  g  k,  pf  x'  at  right 
angles  to  the  plane  A  C,  and 
consequently  parallel  to  each 
other  ;  join  x  x',  and  since  the 
points  p,  g,  p',  are  in  a  straight 
line,  the  points  x,  k,  x'  will  also 
be  in  a  straight  line,  and  there- 
fore x  x'  will  pass  through  k. 
Join  g  p"  ,  and  let  G  be  the  com- 
mon centre  of  gravity  ofp,p',p'f; 
draw  G  K,  p"  x",  perpendicular 
to  the  plane  ;  and  through  g 
draw  m  n  parallel  to  x  x'  meet- 
ing p  x  produced  in  n. 

Now  p  :  p'  ::  p'  g  :  p  g  :  :  (by  sim.  triangles)  p'  mipn; 

.'.  p  fc  p  n  —  p'  x  p'  m,  or  p  x  (n  x  —  p  x)  =  p'  x  (p1  x'  —  m  x')  ; 
but 

n  x  =  g  k  =  m  x',  .'.  p  x  (g  k  —  p  x)  =  p'  x  (p'  x'  —  g  Ic), 
and 


\ 


for  the  same  reason,  if  p  -f-  p  is  placed  at  g,  we  have 


r  K-  (P+P')xff*+P"*P"x"  -P  xpv+P1  XP'X'+P"XP"X" 
'  '"  P+P'+P" 


a  formula  which  is  applicable  to  any  number  of  bodies. 

Let  the  last  equation  be  multiplied  by  the  denominator  of  the 
fraction,  and  we  have 
(p  +  p'+p"+  &c.)  GK=pxpx+p'  xp'  x'+p"xp"x"  +  &c.: 

that  is,  the  moment  of  any  system  of  bodies  with  reference  to  a  given 
plane,  equals  the  sum  of  the  moments  of  all  the  parts  of  the  system 
witli  reference  to  the  same  plane. 

79.  Centre  of  Gravity  of  a  Trapezoid.  —  As  an  example 
of  the  foregoing  principle,  let  it  be  proposed  to  find  the  centre  of 
4 


50 


MECHANICS. 


FIG.  45. 


gravity  of  a  trapezoid,  considered  as  composed  of  two  triangles. 

The  centre  of  gravity  of  the  trapezoid  A  C  (Fig.  45)  is  in  E  F, 

which  bisects  all  the  lines  of  the 

figure  parallel  to  B  C.     Suppose 

G  to  be  the  centre  of  gravity  of 

the   trapezoid ;  through    G   draw 

K  M  perpendicular  to  the  bases. 

Let  KM=.  h,BC—B,AD  —  b, 

and  join  B  D. 

The  moment  of  the  trapezoid 
with  reference  to  B  (7  is 

(B  +  b)  -  •  G  K. 

The  moment  of  the  upper  triangle  is  -^-  •  -  h ;  the  moment  of  the 

Bh    h 
lower  triangle  is  — -  •  5  ; 

<i      •> 


B 


—  •  -  h  ;  whence 

6          O 

£  -\-  2  b 


-      . 

£>  +  0         o 

By  similar  triangles 


the  centre  of  gravity  of  a  trapezoid  is  on  the  line  which  bisects  the 
parallel  bases,  and  divides  it  in  the  ratio  of  twice  the  longer  plus 
the  shorter  to  twice  the  shorter  plus  the  longer. 

1.  Four  bodies,  A,  B,  C,  D,  weighing,  respectively,  2,  3,  6, 
and  8  pounds  are  placed  with  their  centres  of  gravity  in  a  right 
line,  at  the  distance  of  3,  5,  7,  and  9  feet  from  a  given  point  ; 
what  is  the  distance  of  their  common  centre  of  gravity  from  that 
given  point  ;  and  between  which  two  of  the  bodies  does  it  lie  ? 

Ans.  Between  (7  and  Z>;  and  its  distance  from  the  given 
point  7^  feet. 

2.  There  are  five  bodies,  weighing,  respectively,  1,  14,  21£,  22. 
and  29  J  pounds  ;   a  plane  is  assumed  passing  through  the  last 
body,  and  the  distances  of  the  other  four  from  the  plane  are,  re- 
spectively, 21,  5,  6,  and  10  feet  ;  how  far  from  the  plane  is  the 
common  centre  of  gravity  of  the  five  bodies?  Ans.  5  feet.. 

80.   Centrobaric   Mensuration.  —  The    properties  of  the 
centre  of  gravity  furnish  a  very  simple  method  of  measuring 


CENTROBARIC    MENSURATION.  51 

surfaces  and  solids  of  revolution.  This  method  is  comprehended 
in  the  two  following  propositions,  known  as  the  theorems  of 
Guldinus  : 

1.  If  any  line  revolve  about  a  fixed  axis,  which  is  in  the  plane 
of  that  line,  the  SURFACE  which  it  generates  is  equal  to  the  product 
of  the  given  line  into  the  circumference  described  by  its  centre  of 
gravity. 

Let  any  line,  either  straight  or  curved,  revolve  about  a  fixed 
axis  which  is  in  the  plane  of  that  line  ;  and  let/,  /',  /",/"',  etc., 
denote  elementary  portions  of  the  line,  d,  d',  d",  d'",  &c.,  the 
distances  of  these  portions,  respectively,  from  the  axis  ;  then  the 
surface  generated  by/,  in  one  revolution,  will  be  2  TT  df\  hence 
the  surface  generated  by  the  whole  line  will  be 

S  =  2  TT  (df+d'f  +  d"f"  +  d'"f"  '+  &c.)  .  .  .  (1). 

Put  L  =  the  length  of  the  revolving  line,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  the  line;  then 
(Art.  78) 

G  L  =  df+d'f  +  d"f"  +  d'"f'"  +  &c..  .  .  (2). 
Combining  (1)  and  (2),  we  have 

S=2nG  L    .........  (3). 

2.  If  a  plane  surface,  of  any  form  whatever,  revolve  about  a  fixed 
axis  which  is  in  its  own  plane,  the  VOLUME  generated  is  equal  to 
the  product  of  that  surface  into  the  circumference  described  by  its 
centre  of  gravity. 

Let  any  plane  surface  revolve  about  an  axis  which  is  in  the 
plane  of  that  surface  ;  and  let/,/',  /",/"',  &c.,  denote  elementary 
portions  of  the  surface,  d,  d',  d",  d'",  &c.,  the  distances  of  these 
portions,  respectively,  from  the  axis  ;  then  the  volume  generated 
by  /in  one  revolution  will  be  2  TT  d  f  \  hence  the  volume  generated 
by  the  whole  surface  will  be 


dff'  +  d"f"  +  d'"f'"  +  Ac.)  .  .  (4). 

Put  A  =  the  area  of  the  revolving  surface,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  that  surface  ;  then 
(Art.  78) 

A  G  =  df  +  d'f  +  d"f"  +  d'"f"  +  &c.,  .  .  .  (5). 
Substituting  in  (4),  we  have 

V=27rAG  .........   (6). 

As  an  illustration  of  the  first  theorem,  the  straight  line  CD 
(Fig.  46),  revolving  about  the  center  G,  describes  a  circle  whose 


MECHANICS. 


FIG.  46. 


surface  is  equal  to  C  D  into  the  circumference  of  the  circle  de- 
scribed by  its  centre  of  gravity,  E.  This  is  evident  also  from  the 
consideration  that,  since  E  is  the  centre  of  the  line  C  D,  the  cir- 
cumference described  by  it  will  be  half  the 
length  of  the  circumference  A  D  B  ;  and  the 
area  of  a  circle  is  equal  to  the  product  of  the 
radius  into  half  the  circumference. 

The  second  theorem  is  illustrated  by  the 
volume  of  a  cylinder,  whose  height  =  h,  and 
the  radius  of  whose  base  =  r. 

Common  method  ;  base=7r  r2 ;  height=&; 
.*.  vol.  =  TT  r2  h. 

Centrobaric  method ;  revolving  area  =  r  h  ;  circumference 
described  by  the  centre  of  gravity  =  £  r  x  2  TT  ;  /.  vol.  =  r  h  . 
£  r  .  2  TT  =  TT  r*  h. 

81.  Examples.— 

1.  Suppose  the  small  circle  (Fig.  46)  to  be  placed  with  its 
plane  perpendicular  to  the  plane  of  the  paper,  and  revolved  about 
C,  the  point  D  describing  the  line  D  B  A  ;  required  the  content 
of  the  solid  ring.     If  C  D  =  R,  and  E  D  =  r,  then  the  area  re- 
volved =  TT  r2,  and  the  circumference  D  B  A  =  2  TT  R  ;  .*.  the  ring 
=  2  TT2  R  r2.     It  is  equal  to  a  cylinder  whose  base  is  the  circle 
ED,  and  whose  height  equals  the  line  DBA. 

2.  Find  the  convex  surface  of  a  cone  ;  slant  height  =  s  ;  and 
rad.  of  base  =  r.     The  line  revolved  being  s,  and  the  distance 
from  the  axis  to  its  centre  of  gravity,  £  r,  the  surface  is  TT  r  s. 

3.  A  square,  whose  side  is  one  foot,  is  revolved  about  an  axis 
which  passes  through  one  of  its  angles,  and  is  parallel  to  a  diago- 
nal ;  required  the  volume  of  the  figure  thus  formed. 

Ans.  TT  V%,  or  4.4429  cubic  ft. 

4.  Find  the  centre  of  gravity  of  a  semi-circumference.     In 
this  case   the   revolving   semi-circumference   ABO  (Fig.  47) 
generates  the  surface  of  a  sphere ;  hence,  taking  the 
diameter  A  C  as  an  axis,  calling  the  distance  of  the 

centre  of  gravity  G  from  the  axis  x  and  radius  r,  we 
have 

2r 


FIG.  47. 


4  7T  r2  =  2  TT  #  x  TT  r 


a?  = 


Hence  the  distance  of  the  centre  of  gravity  of  the  semi- 
circumference  from  the  centre  of  the  circle  is 


2  r 


=  .637  r. 


5.  Find  the  centre  of  gravity  of  a  semicircle.    The  revolving 


DIFFERENT    KINDS    OF    EQUILIBRIUM.  53 

area  generates  a  sphere,  and  hence,  as  in  the  preceding  problem, 
we  have 

4  r 
%nrs  =  2'xxx:k'Kr*',   .'.  x  =  ~  —  =  .424  r. 

In  any  case,  when  a  simple  expression  for  the  surface  generated 
by  a  revolving  line  can  be  found,  it  is  easy  to  find  the  centre  of 
gravity  of  the  line  by  this  method,  and  the  centre  of  gravity  of 
an  area  may  be  readily  found  from  the  expression  of  the  volume 
generated. 

82.  Support  of  a  Body. — A  body  cannot  rest  on  a  smooth 
plane  unless  it  is  horizontal ;  for  the  pressure  on  a  plane  (Art.  G5) 
cannot  be  balanced  by  the  resistance  of  that  plane,  except  when 
perpendicular  to  it ;  therefore,  as  the  force  of  gravity  is  vertical, 
the  resisting  plane  must  be  horizontal. 

The  base  of  support  is  that  area  on  the  horizontal  plane  which 
is  contprehended  by  lines  joining  the  extreme  points  of  contact. 

If  there  are  three  points  of  contact,  the  base  is  a  triangle  ;  if 
four,  a  quadrilateral,  &c. 

When  the  vertical  through  the  centre  of  gravity  (called  the 
line  of  direction)  falls  within  the  base,  the  body  is  supported  ;  if 
without,  it  is  not  supported.  In  the  body  A  (Fig.  48)  the  force 
of  gravity  acts  in  the  line 

G  F,  and  there  are  lines  of  FlG'  4a 

resistance  on  both  sides  of 
G  F,  as  G  O  and  G  E,  so 
that  the  body  cannot  turn 
on  the  edge  of  the  base, 
without  rising  in  an  arc 
whose  radius  is  G  C  or  G  E. 
But,  in  the  body  B,  there  is  resistance  only  on  one  side ;  and 
therefore,  if  the  force  of  gravity  be  resolved  on  G  C  and  a  perpen- 
dicular to  it,  the  body  is  not  prevented  from  moving  in  the  direc- 
tion of  the  latter,  that  is,  in  the  arc  whose  radius  is  G  C. 

If  the  line  of  direction  fall  at  the  edge  of  the  base,  the  least 
force  will  overturn  it. 

83.  Different  Kinds  of  Equilibrium.— If  the  base  is  re- 
duced to  a  line  or  point,  then,  though  there  may  be  support,  there 
is  no  firmness  of  support ;  the  body  will  be  moved  by  the  least 
force.  But  it  is  affected  very  differently  in  different  cases. 

When  it  is  moved  from  its  position  of  support  and  left,  it  will 
in  some  cases  return  to  it,  pass  by,  and  return  again,  and  continue 
thus  to  vibrate  till  it  settles  in  its  place  of  support  by  friction  and 
other  resistances.  This  condition  is  called  stable  equilibrium. 


MECHANICS. 


In  other  cases,  when  moved  from  its  position  of  support  and 
left,  it  will  depart  further  from  it,  and  never  recover  that  position 
again.  This  is  called  unstable  equilibrium. 

In  other  cases  still,  the  body,  when  moved  from  its  place  of 
support  and  left,  will  remain,  neither  returning  to  it  nor  depart- 
ing further  from  it.  This  is  called  neutral  equilibrium. 

84.  Stable  Equilibrium.— Let  the  body  (Fig.  49)  be  sus- 
pended on  the  pivot  A.     This  is  its  base  of  support.     While  the 
centre  of  gravity  is  below  A,  the  line  of  direction 

EOF  passes  through  the  base,  and  the  body  is 

supported.     Let  it  be  moved  aside,  and  the  centre 

of  gravity  be  left  at  G.     Let  G  R  represent  the 

force  of  gravity,  and  resolve  it  into  G  N  on  the 

line  A  G,  and  N  R,  or  G  B,  perpendicular  to  A  G. 

G  N  is  resisted  by  the  strength  of  A,  and  G  B 

moves  the  centre   of  gravity  in  the  arc  whose 

radius  is  A  G.     Hence  the   body  swings  with 

accelerated  motion  till  the  centre  of  gravity  reaches 

0,  where  the  force  G  B  becomes  zero.     But  by 

its  inertia,  the  body  passes  beyond  that  position, 

and  ascends  on,  the  other  side,  till  the  retarding 

force  of  gravity  stops  it  at  g,  as  far  from  0  as  G  is. 

It  then  descends  again,  and  would  never  cease  to  oscillate  were 

there  no  obstructions. 

85.  Unstable  Equilibrium. — Next,  let  the  body  be  turned 
on  the  pivot  till  the  centre  of  gravity  G  is  at  P,  above  A  (Fig.  50). 
Then,  as  well  as  when  G  is  below  A,  the  body  is 
supported,   because   the   line   of  direction  E  P  F 

passes  through  the  base  A.  But  if  turned  and  left 
in  the  slightest  degree  out  of  that  position,  it  can- 
not recover  it  again,  but  will  depart  further  and 
further  from  it.  Let  G  R  represent  the  force  of 
gravity,  and  let  it  be  resolved  into  G  N,  acting 
through  A,  and  G  B  perpendicular  to  it.  The 
former  is  resisted  by  A  ;  the  latter  moves  G  away 
from  P,  the  place  of  support.  If  the  body  is  free 
to  revolve  about  A,  without  falling  from  it,  the 
centre  of  gravity  will,  by  friction  and  other  resist- 
ances, finally  settle  below  A,  as  in  the  case  of 
stable  equilibrium. 

86.  Neutral  Equilibrium. — Once  more,  suppose  the  pivot 
supporting  the  body  to  be  at  G,  the  centre  of  gravity ;  then,  in 


FIG.  50. 


CENTRE   OF    GRAVITY.  55 

whatever  situation  the  body  is  left,  the  line  of  direction  passes 
through  the  base,  and  the  body  rests  indifferently  in  any  position. 

These  three  kinds  of  equilibrium  may  be  illustrated  also  by 
bodies  resting  by  curved  surfaces  on  a  horizontal  plane.  Thus,  if 
a  cylinder  is  uniformly  dense,  it  will  always  have  a  neutral  equi- 
librium, remaining  wherever  it  is  placed.  But  if,  on  account  of 
unequal  density,  its  centre  of  gravity  is  not  in  the  axis,  then  its 
equilibrium  is  stable,  when  the  centre  of  gravity 
is  below  the  axis,  and  unstable  when  above  it. 

In  general,  there  is  stable  equilibrium  when 
the  centre  of  gravity,  on  being  disturbed  in  either 
direction,  begins  to  rise ;  unstable  when,  if  dis- 
turbed either  way,  it  begins  to  descend;  and 
neutral  when  the  disturbance  neither  raises  nor 
lowers  the  centre  of  gravity. 

87.  Questions  on  the  Centre  of  Grav- 
ity.- 

1.  A  frame  20  feet  high,  and  4  feet  in  diam- 
eter, is  racked  into  an  oblique  form  (Fig.  51), 
till  it  is  on  the  point  of  falling ;   what  is  its 
inclination  to  the  horizon  ?  Ans.  78°  27'  47". 

2.  A  stone  tower,  of  the  same  dimensions  as  the  former,  is  in- 
clined till  it  is  about  to  fall,  but  preserves  its  rectangular  form  ; 
what  is  its  inclination  ?  Ans.  78°  41'  24". 

3.  A  cube  of  uniform  density  lies  on  an  inclined  plane,  and  is 
prevented  by  friction  from  sliding  down  ;   to  what  inclination 
must  the  plane  be  tipped,  that  the  cube  may  just  begin  to  roll 
down  ?  Ans.  45°. 

4.  What  must  be  the  inclination  of  a  plane,  in  order  that  a 
regular  prism  of  any  given  number  of  sides  may  be  on  the  point 
of  rolling  down,  if  friction  prevents  sliding  ? 

Ans.  Equal  to  half  the  angle  at  the  centre  of  the  prism, 
subtended  by  one  side. 

5.  A  body  weighing  83  Ibs.  is  suspended,  and  drawn  aside  from 
the  vertical  9°  by  a  horizontal  force ;  what  pressure  is  there  on 
the  point  of  support,  and  what  force  urges  it  down  the  arc  ? 

Ans.  Pressure  on  the  support,  81.978  Ibs. 
Moving  force,  12. 984  Ibs. 

88.  Motion  of  the  Centre  of  Gravity  of  a  System  when 
one  of  the  Bodies  is  Moved. — 

WJien  one  body  of  a  system  is  moved,  the  centre  of  gravity  of 
the  system  moves  in  a  similar  path,  and  its -velocity  is  to  that  of 


56  MECHANICS. 

the  moving  body  as  the  mass  of  that  body  is  to  the  mass  of  the  whole 
system. 

If  the  system  contains  but  two  bodies,  A  and  B  (Fig.  52),  sup- 
pose A  to  remain  at  rest,  while  B 
describes  the  straight  lines  B  C, 
C  D,  &c.,  the  centre  of  gravity  G 
will  in  the  same  time  describe  the 
similar  series,  G  H,  H  J,  &c. 
When  B  is  in  the  position  B,  and 
the  centre  of  gravity  at  G,  A  G  : 
A  B  :  :  B  :  A  +  B  ;  when  B  is 
at  C,  A  H  :  A  C  :  :  B  :  A  +  B ; 
.'.AG'.AB'.-.AH'.AC.  Hence 
G  H  is  parallel  to  B  C,  and  G  H  :  B  C : :  B  :  A  +  B.  In  like 
manner,  H  J  :  C  D  : :  B  :  A  -f-^,  &c.  Thus  all  the  parts  of  one 
path  are  parallel  to  the  corresponding  parts  of  the  other,  and  have 
a  constant  ratio  to  them.  Therefore  the  paths  are  similar.  As 
the  corresponding  parts  are  described  in  equal  times,  their  lengths 
are  as  the  velocities.  But  the  lengths  are  as  B  :  A  -f-  B ;  there- 
fore the  velocity  of  the  common  centre  of  gravity  is  to  that  of  the 
moving  body  as  the  mass  of  the  moving  body  is  to  the  mass  of  both. 
The  same  reasoning  is  applicable  when  the  body  moves  in  a  curve. 

If  the  system  contain  any  number  of  bodies,  and  the  centre  of 
gravity  of  the  whole  be  at  G,  then  the  centre  of  gravity  of  all 
except  B  must  be  in  the  line  B  G  beyond  G.  Suppose  it  to  be  at 
A9  and  to  remain  at  rest,  while  B  moves  ;  then  it  is  proved  in  the 
same  manner  as  before,  that  G,  the  centre  of  gravity  of  the  whole 
system,  moves  in  a  path  parallel  to  the  path  of  B,  and  with  a 
velocity  which  is  to  B's  velocity  as  the  mass  of  B  to  the  mass  of 
the  entire  system. 

89.  Motion  of  the  Centre  of  Gravity  of  a  System  when 
Several  of  the  Bodies  are  Moved.— 

When  any  or  all  of  the  bodies  of  a  system  are  moved,  the  centre 
of  gravity  moves  in  the  same  manner  as  if  all  the  system  were 
collected  there,  and  acted  on  by  the  forces  ivliich  act  on  the  separate 
bodies. 

Let  A,  B,  (7,  &c.  (Fig.  53),  belong  to  a  system  containing  any 
number  of  bodies,  and  let  M  be  the  mass  of  the  system.  Let  A 
be  moved  over  A  a,  B  over  B  b,  G  over  C  c,  &c.  And  first  sup- 
pose the  motions  to  be  made  in  equal  successive  times.  If  the 
centre  of  gravity  of  the  system  is  first  at  G,  then  that  of  all  the 
bodies  except  A  is  in  A  G  produced,  as  at  g.  While  A  moves  to  #, 
G  moves  in  a  parallel  line  to  H  (Art.  88),  and  G  H:  A  a  ::  A  :  M. 
In  like  manner,  when  B  describes  B  b,  the  centre  of  gravity  of  the 
other  bodies  being  at  h,  the  centre  of  gravity  of  the  system  de- 


MOTION    OF    CENTRE    OF    GRAVITY. 


57 


FIG.  53. 


scribes  the  parallel  line,  H  K,  and  H  K  :  B  b  : :  B  :  M ;  and  when 
C  moves,  K  L  :  C c  ::  C :  M,  &c.  Now,  A  a  and  G  H  represent 
the  respective  velocities  of  the 
body  Ay  and  the  system  M\ 
therefore,  if  we  convert  the 
proportion  G  H  :  A  a  : :  A  :  M 
into  an  equation,  we  have  A  x 
A  a  =  M  x  G  H]  that  is,  the 
momentum  of  the  body  A 
equals  the  momentum  of  the 
system  M.  It  therefore  re- 
quires the  same  force  to  move 

A  over  A  a  as  to  move  the  system  M  over  G  H.  The  same  is  true 
of  the  other  bodies.  If  then  the  several  forces  which  move  the 
bodies,  limiting  the  number  to  three,  for  the  present,  were  applied 
successively  to  the  system  collected  at  G,  they  would  move  it  over 
G  H,  H  K,  K  L.  But  if  applied  at  once,  they  would  move  it  over 
G  L,  the  remaining  side  of  the  polygon.  If,  therefore,  the  forces, 
instead  of  acting  successively  on  the  bodies,  were  to  move  A  over 
A  a,  B  over  B  Z>,  and  G  over  C  c,  at  the  same  time,  the  centre  of 
gravity  of  the  system  would  describe  G  L  in  the  same  time.  In  the 
same  way  it  may  be  proved,  that  whatever  forces  are  applied  to  the 
several  bodies  of  a  system,  the  centre  of  gravity  of  the  system  is 
moved  in  the  same  manner  as  a  body  equal  to  the  whole  system 
would  be  moved,  if  all  the  same  forces  were  applied  to  it. 

It  is  possible  that  the  centre  of  gravity  of  a  system  should 
remain  at  rest,  while  all  the  bodies  in  it  are  in  motion.  For,  sup- 
pose all  the  forces  acting  on  the  bodies  to  be  such  that  they  might 
be  represented  in  direction  and  intensity  by  all  the  sides  of  a  poly- 
gon, then,  since  a  single  body  acted  on  by  them  would  be  in  equi- 
librium, therefore  the  centre  of  gravity  of  the  system  would  remain 
at  rest,  though  the  bodies  composing  it  are  in  motion. 

90.  Mutual  Action  among  the  Bodies  of  a  System. — 

The  forces  which  have  been  supposed  to  act  on  the  several  bodies 
of  a  system  are  from  without,  and  not  forces  which  some  of  the 
bodies  within  the  system  exert  on  others.  If  the  bodies  of  a  sys- 
tem mutually  attract  or  repel  each  other,  such  action  cannot  affect 
the  centre  of  gravity  of  the  whole  system.  For  action  and  reac- 
tion are  always  opposite  and  equal.  Whatever  force  one  body 
exerts  on  any  other  to  move  it,  that  other  exerts  an  equal  force  on 
the  first,  and  the  two  actions  produce  equal  and  opposite  effects 
on  the  centre  of  gravity  between  them.  Therefore  the  centre  of 
gravity  of  a  system  remains  at  rest,  if  the  bodies  which  compose 
it  are  acted  on  only  by  their  mutual  attractions  or  repulsions. 

91.  Examples  on  the  Motion  of  the  Centre  of  Gravity.— 


58 


MECHANICS. 


FIG.  54. 


FIG,  55, 


1.  Two  bodies,  A  and  B,  of  given  weights,  start  together  from 
D  (Fig.  54),  and  move  uniformly  with  given  velocities  in  the  direc- 
tions D  A  and  D  B ;  required  the  di- 
rection and  velocity  of  their  centre  of 

gravity. 

As  the  directions  of  D  A  and  D  B 
are  given,  we  know  the  angle  ADS; 
from  the  given  velocities,  we  also  know 
the  lines  D  A  and  D  B,  described  in  a 
certain  time.  Calculate  the  side  A  B, 
and  the  angles  A  and  B.  Find  the 

place  of  the  centre  of  gravity  O  between  the  bodies  at  A  and  B. 
Then,  in  the  triangle  D  B  G,  D  B,  B  G,  and  angle  B  are  known, 
by  which  may  be  found  the  distance  D  G  passed  over  by  the 
centre  of  gravity  in  the  time,  and  B  D  G  the  angle  which  its  path 
makes  with  that  of  the  body  B. 

2.  Three  bodies  of  given  weight,  A,  B,  C,  in  the  same  time 
and  in  the  same  order,  describe  with  uniform  velocity  the  three 
sides  of  the   given    triangle 

ABC  (Fig.  55)  ;   required  A 

the  path  of  their  centre   of 

gravity. 

Let  G  be  their  centre  of 
gravity  before  they  move.  If 
they  move  successively,  G  de- 
scribes G  K>  K  L,  L  M,  par- 
allel to  the  sides  of  the  trian- 
gle, and  having  to  them  re- 
spectively the  same  ratios  as 
the  corresponding  moving  bodies  have  to  the  sum  of  the  bodies 
(Art.  89).  Thus,  three  sides  of  the  polygon  are  known  ;  and  the 
angle  K  =  B,  and  L  =  C.  These  data  are  sufficient  for  calcu- 
lating the  fourth  side,  G  M,  which  the  centre  of  gravity  describes, 
when  the  bodies  move  together. 

3.  Show  that  when  the  three  bodies  in  Example  2  are  equal, 
the  centre  of  gravity  will  remain  at  rest. 

4.  A  (Fig.  56)  weighs  one  pound  ; 

B  weighs  two  pounds,  and  lies  direct-  22 
ly  east  of  A  ;  they  move  simulta- 
neously, A  northward,  and  B  east- 
ward, at  the  same  uniform  rate  of  40 
feet  per  second  ;  required  the  direc- 
tion and  velocity  of  their  centre  of 

gravity. 

Ans.  Velocity  is  29.814  feet  per  second. 
Direction  is  E.  26°  33'  54"  N. 


CHAPTER  V. 


FIG.  57. 


THE  COLLISION   OF   BODIES. 

92.  Elastic  and   Inelastic    Bodies.— Mastic  bodies  are 
those  which,  when  compressed,  or  in  any  way  altered  in  form, 
tend  to  return  to  their  original  state.     Those  which  show  no  such 
tendency  are  called  inelastic  or  non-elastic.    No  substance  is 
known  which  is  entirely  destitute  of  the  property  of  elasticity ; 
but  some  have  it  in  so  small  a  degree,  that  they  are  called  in- 
elastic, such  as  lead  and  clay.      Elasticity  is  perfect  when  the 
restoring  force,  whether  great  or  small,  is  equal  to  the  compress- 
ing force.     Air,  and  the  gases  generally,  seem  to  be  almost  per- 
fectly elastic ;  ivory,  glass,  and  tempered  steel,  are  imperfectly, 
though  highly,  elastic ;  and  in  different  substances,  the  property 
exists  in  all  conceivable  degrees  between  the  above-named  limits. 

93.  Mode  of  Experimenting.— Experiments  on  collision 
may  be  made  with  balls  of  the 

same  density  suspended  by 
long  threads,  so  as  to  move  in 
the  line  which  joins  their  cen- 
tres of  gravity.  If  the  arcs 
through  which  they  swing  are 
short  compared  with  their 
radii,  the  balls,  let  fall  from 
different  heights,  will  reach  the 
bottom  sensibly  at  the  same 
time,  and  will  impinge  with 
velocities  which  are  very  nearly 
proportional  to  the  arcs.  Thus 
A  (Fig.  57),  falling  from  6, 
and  B  from  3,  will  come  into  collision  at  0,  with  velocities  which 
are  as  2  :  1. 

94.  Collision  of  Inelastic  Bodies. — Such  bodies,  after  im- 
pact, move  together  as  one  mass. 

The  velocity  of  two  inelastic  bodies  after  collision  is  equal  to  the 
algebraic  sum  of  their  momenta,  divided  by  the  sum  of  the  bodies. 

Let  Ay  B,  represent  the  masses  of  the  two  bodies,  and  #,  #, 
their  respective  velocities.  Considering  a  as  positive,  if  B  moves 


60  MECHANICS. 

in  the  opposite  direction,  its  velocity  must  be  called  —  b.  Let  v 
be  the  common  velocity  after  impact,  and  suppose  the  bodies  to 
be  moving  in  the  same  direction,  the  momentum  of  A  is  A  a  ; 
that  of  B  is  B  b  ;  and  the  momentum  of  both  after  collision  is 
(A  -f  B)  v.  According  to  the  third  law  of  motion  (Art.  13), 
whatever  momentum  A  loses,  B  gains,  so  that  the  whole  momen- 
tum is  the  same  after  collision  as  before  ;  therefore 

A  a  +  B  b  =  (A  +  B)  v9  .'.  v  =^±±-L*. 


To  find  the  gain  or  loss  of  velocity  of  either  body  subtract  the 
resulting  velocity  from  the  original  velocity  ;  a  negative  result 
indicates  motion  in  a  direction  opposite  to  the  original  motion. 

95.  Questions  on  Inelastic  Bodies.— 

1.  A}  weighing  3  oz.,  and  moving  10  feet  per  second,  overtakes 
B,  weighing  2  oz.,  and  moving  3  feet  per  second  ;  what  is  the  com- 
mon velocity  after  impact  ?  Ans.  7-J-  feet  per  second. 

2.  A  weight  of  7  oz.,  moving  11  feet  per  second,  strikes  upon 
another  at   rest   weighing   15   oz.  ;   required   the   velocity   after 
impact  ?  Ans.  3£  feet  per  second. 

3.  A  weighs  4  and  B  2  pounds  ;  they  meet  in  opposite  direc- 
tions, A  with  a  velocity  of  9,  and  B  with  one  of  5  feet  per 
second  ;  what  is  the  common  velocity  after  impact  ? 

Ans.  4J  feet  per  second. 

4.  A  =  7  pounds,  B  =  4  pounds  ;    they  move  in  the  same 
direction,  with  velocities  of  9  and  2  feet  per  second  ;  required  the 
velocity  lost  by  A  and  gained  by  B?  Ans.  A  2-fa,  B  4/T. 

5.  A  body  moving  7  feet  per  second,  meets  another  moving  3 
feet  per  second,  and  thus  loses  half  its  momentum  ;   what  are 
the  relative  masses  of  the  two  bodies  ? 

Ans.    A  :  B  ::  13  :  7. 

6.  A  weighs  6  pounds  and  B  5  ;  B  is  moving  7  feet  per  second 
in  the  same  direction  as  A  ;  by  collision  B's  velocity  is  doubled  ; 
what  was  A's  velocity  before  impact? 

Ans.  19f  feet  per  second. 

7.  A  body  weighing  100  Ibs.,  and  having  velocity  40  feet  per 
second  meets  another  weighing  20  Ibs.,  and  having  velocity  of  200 
feet  per  second  ;  what  will  be  the  velocity  after  impact  ? 

Ans.  0. 

96.  Collision  of  Elastic  Bodies.  —  Elastic  bodies  after  col- 
lision do  not  move  together,  but  each  has  its  own  velocity.    These 
velocities  are  found  by  doubling  the  loss  and  gain  of  inelastic 
bodies.     When  the  elastic  body  A  impinges  on  B,  it  loses  velocity 


QUESTIONS    ON    ELASTIC    BODIES.  61 

while  it  is  becoming  compressed,  and  again,  while  recovering  its 
form,  it  loses  as  much  more,  because  the  restoring  force  is  equal  to 
the  compressing  force.  For  a  like  reason,  B  gains  as  much  velo- 
city while  recovering  its  form  as  it  gained  while  being  compressed 
by  the  action  of  A.  Hence,  doubling  the  expressions  for  loss  and 
gain  found  by  Art.  94,  and  applying  them  to  the  original  veloci- 
ties, we  find  the  velocity  of  each  body  after  collision,  on  the  sup- 
position of  perfect  elasticity. 

Two  equal  elastic  bodies,  A  and  B,  weighing  50  Ibs.  each, 
moving  with  velocities,  A  =  40  ft.,  and  B  =  20  ft.  per  second, 
meet;  what  will  be  the  velocity  of  each  after  impact?  First  we 
must  find  the  gain  and  loss  of  velocity  on  the  supposition  that 
the  bodies  are  inelastic,  and  then  double  such  gain  or  loss  ;  there- 
fore, according  to  Art.  94,  calling  the  velocity  after  impact  v,  we 
have  (50  +  50)  v  =  50  x  40  —  50  x  20,  calling  the  velocity  of 
B  negative  as  the  bodies  move  in  opposite  directions, — 

1000 
whence  v  =  —--=  10. 

A  loses  40  —  10  =  30  ft.  per  second,  and  B  loses  —  20  — 
10  =  —  30  feet  per  second  ;  that  is  to  say,  B  loses  all  its  motion 
in  its  original  direction,  and  moves,  backward  with  velocity  10. 

Now  as  these  are  elastic  we  must  double  the  gain  and  loss,  and 
we  have  A's  loss  =  60  ft.  and  B's  =  —  60  ft. ;  therefore  A  must 
move  with  velocity  40  —  60  =  —  20,  and  B  with  velocity  —  20 
—  (—  60)  =  40,  hence  A  must  now  move  in  a  direction  opposite 
to  the  first,  with  velocity  20,  and  B  also  in  direction  opposite  to 
its  previous  motion,  with  velocity  40.  Each  body  takes  the  velocity 
of  the  other  when  the  bodies  are  equal. 

97.  Questions  on  Elastic  Bodies. — 

1.  A,  weighing  10  Ibs.  and  moving  8  feet  per  second,  impinges 
on  B,  weighing  6  Ibs.  and  moving  in  the  same  direction,  5  feet 
per  second  ;  what  are  the  velocities  of  A  and  B  after  impact  ? 

Ans.  A's  =  5f,  B's  =  8}. 

2.  A  :  B  : :  4  :  3  ;  directions  the  same ;  velocities  5:4;  what 
is  the  ratio  of  their  velocities  after  impact  ?  Ans.  29  :  36. 

3.  A,  weighing  4  Ibs.,  velocity  6,  meets  B,  weighing  8  Ibs., 
velocity  4  ;  required  their  respective  directions  and  velocities  after 
collision  ?  Ans.  A  is  reflected  back  with  a  velocity  of  ?|, 

and  B  with  a  velocity  of  2f . 

4.  A  and  B  move  in  opposite  directions ;  A  equals  4  B,  and 
"b  =  2  a  ;  how  do  the  bodies  move  after  collision  ? 

Ans.  A  returns  with  \,  B  with  If  its  original  velocity. 


MECHANICS. 


FIG.  58. 


98.  Series  of  Elastic  Bodies.— 

1.  Equal  bodies.— Let  a  row  of  equal  elastic  bodies,  A,  B,  C. . .  P 
(Fig.  58)  be  suspended  in  contact ;   then 

(Art.  96),  if  A  be  drawn  back  and  left  to 
fall  against  B,  it  will  rest  after  impact,  and 
B  will  tend  to  move  on  with  A's  velocity; 
after  the  impact  of  B  on  (7,  B  will  remain, 
and  C  tend  to  move  with  the  same  velocity  ; 
and  so  the  motion  will  be  transmitted  through 
the  series,  and  F  will  move  away,  while  all 
the  others  remain  at  rest. 

2.  Decreasing  series. — If  the  bodies  de- 
crease, as  A,  By  C,  &c.  (Fig.  59),  and  A  be 
drawn  back  to  A',  and  allowed  to  fall  against 
B,  then  A  still  moves  forward,  while  B  re- 
ceives a  greater  velocity  than  A  had,  C  still 

greater,  &c.  The  last  of  the  series,  therefore,  moves  with  the 
greatest  velocity,  and  each  one  with  a  greater  velocity  than  that 
which  impinged  on  it. 

FIG.  59. 


3.  Increasing  series. — If  the  bodies  increase,  as  A,  B,  C,  &o. 
(Fig.  60),  then,  when  A  falls  from  A'  against  B,  it  imparts  to  B  a 


FIG.  60. 


velocity  than  it  had  itself,  and  rebounds  ;  in  like  manner  B 
rebounds  from  (7,  and  so  on ;  while  the  last  of  the  series  goes  for- 
ward with  less  velocity  than  any  previous  one  would  have  had  if 
it  had  been  the  last. 

1.  There  are  ten  bodies  whose  masses  increase  geometrically 
by  the  constant  ratio  3,  and  the  first  impinges  on  the  second  with 
the  velocity  of  5  feet  per  second ;  required  the  motion  of  the  last 
body?  Ans.  The  last  body  would  move  with  the 

velocity  of  -$-%  feet  per  second. 


LOSS    OF    LIVING    FORCE.  63 

99.  Living  Force  lost  in  the  collision  of  Inelastic 
Bodies.  —  The  amount  of  living  force  (Art.  37)  before  collision  is 

A  a?+BP',  and  after  collision  it  is  (A  +  B)  x 

—  2    ,    2?    •     Subtract  the  latter  from  the  former,  and  call  the 

remainder  d.    Then  d=A  a?  +  B  l^—^Aa  +  B  ^\     Expanding 

and  uniting  terms,  d=—-  ^rj^-*     This  value  of  d  is  positive, 

because  (a—  #)2is  necessarily  positive,  as  well  as  A  and  B.  There- 
fore there  is  always  a  loss  of  living  force  in  the  collision  of  inelastic 
bodies. 

This  motion,  or  visible  energy,  seemingly  lost,  is  transformed 
into  heat  by  the  impact. 

100.  Living  Force  Preserved  in  the  Collision  of  Elastic 
Bodies.  —  The  living  force  of  A  before  collision  is  A  a2  ;  after 

collision,  it  is  A  x  -*—  ~7^  f^a  --  '  Subtracting  the  latter 
from  the  former,  the  loss  (supposing  there  is  loss)  is 


The  living  force  of  B  before  collision  is  B  &  ;  after  collision,  it 
s       x  "  --  2  >  and  fciie  exPression  for  loss  is 

4:A*Ba* 

Therefore  the  total  loss  of  living  force  is  the  sum  of  the  expres- 
sions (1.)  and  (2.). 

Eeducing  the  two  first  terms  in  each  fraction  to  one,  the  frac- 
tions become 


., 

±AB*tf—±(—  —* 

and 


If  the  fractions  (3)  and  (4)  be  added,  it  is  evident  that  the 
numerators  cancel  each  other,  and  therefore  the  sum  of  the  frac- 
tions is  zero.  Hence,  there  is  no  loss  of  living  force  in  the  col- 
lision of  elastic  bodies.  * 


64 


MECHANICS. 


FIG.  61. 


101.  Impact  on  an  Immovable  Plane.  —  If  an  inelastic 
body  strikes  a  plane  perpendicularly,  its  motion  is  simply  de- 
stroyed; in  strictness,  however,  it  imparts  an  infinitely  small 
velocity  to  the  body  called  immovable.  If  it  strikes  obliquely, 
and  the  plane  is  smooth,  it  slides  along  the  plane  with  a  dimin- 
ished velocity.  Let  A  L  (Fig.  61)  represent  the  motion  of  the 
body  before  impact  on  the  plane 
P  N,  and  resolve  it  into  A  C,  per- 
pendicular,  and  C  L,  parallel  to 
the  plane.  Then  A  C,  as  before, 
is  destroyed,  but  C  L  is  not  af- 
fected ;  hence  the  former  velocity 
is  to  its  velocity  on  the  plane,  as 
A  L  :  C  L  :  :  radius  :  cosine  of  p 
the  inclination. 

If  a  perfectly  elastic  body  impinges  perpendicularly  upon  a 
plane,  then,  after  its  motion  is  destroyed,  the  force  by  which  it 
resumes  its  form  causes  an  equal  motion  in  the  opposite  direction  ; 
that  is,  the  body  rebounds  in  its  own  path  as  swiftly  as  it  struck. 
But  if  the  impact  is  oblique,  the  body  rebounds  at  an  equal  angle 
on  the  opposite  side  of  the  perpendicular.  For,  resolve  A  L,  as 
before,  into  AC,  C  L  ;  the  latter  continues  uniformly  ;  but, 
instead  of  the  component  A  C,  there  is  an  equal  motion  in  the 
opposite  direction.  Therefore,  if  L  D  is  made  equal  to  C  L,  and 
D  E  equal  to  A  <?,  the  resultant  of  L  D  and  D  E  is  L  E,  which 
is  equal  to  A  L,  and  has  the  same  inclination  to  the  plane. 
Hence,  the  angles  of  incidence  and  reflection  are  equal,  and  on 
opposite  sides  of  the  perpendicular  to  the  surface  at  the  place 
of  impact. 

102.  Imperfect  Elasticity.—  The  formulse  for  the  velocity 
of  bodies  after  collision,  and  the  statements  of  the  preceding  arti- 
cle, are  correct  only  on  the  supposition  that  bodies,  are,  on  the 
one  hand,  entirely  destitute  of  elasticity,  or  on  the  other  perfectly 
elastic.  As  no  solid  bodies  are  known,  which  are  strictly  of  either 
class,  these  deductions  are  found  to  be  only  near  approximations 
to  the  results  of  experiment.  In  all  practical  cases  of  the  impact 
of  movable  bodies,  the  loss  and  gain  of  velocity  are  greater  than  if 
they  were  inelastic,  and  less  than  if  perfectly  elastic.  And  in  cases 
of  impact  on  a  plane,  there  is  always  some  velocity  of  rebound, 
but  less  than  the  previous  velocity  ;  and  therefore,  if  the  collision 
is  oblique,  the  body  has  less  velocity,  and  makes  a  smaller  angle 
with  the  plane  than  before.  For,  making  D  F  less  than  A  C,  the 
resultant  L  F  is  less  than  A  L,  and  the  angle  D  L  F  is  smaller 
than  D  L  Ef  or  A  L  C. 


CHAPTER   VI. 

SIMPLE    MACHINES. 

103.  Classification  of  Machines.— In  the  preceding  chap- 
ters, the  motion  of  bodies  has  been  supposed  to  arise  from  the 
immediate  action  of  one  or  more  forces.  But  a  force  may  pro- 
duce effects  indirectly,  by  means  of  something  which  is  inter- 
posed for  the  purpose  of  changing  the  mode  of  action.  These 
intervening  bodies  are  called,  in  general,  machines ;  though  the 
names,  tools,  instruments,  engines,  &c.,  are  used  to  designate  par- 
ticular classes  of  them.  The  elements  of  machinery  are  called 
simple  machines.  The  following  list  embraces  those  in  most 
common  use  : 

1.  The  lever. 

2.  The  wheel  and  axle. 

3.  The  pulley. 

4.  The  rope  machine. 

5.  The  inclined  plane. 

6.  The  wedge. 

7.  The  screw. 

8.  The  knee-joint. 

In  respect  to  principle,  these  eight,  and  all  others,  may  be 
reduced  to  three. 

1.  The  law  of  equal  moments,  applicable  in  those  cases  in 
which  the  machine  turns  on  a  pivot  or  axis,  as  in  the  lever  and 
the  wheel  and  axle. 

2.  The  principle  of  transmitted  tension,  to  be  applied  wherever 
the  force  is  exerted  through  a  flexible  cord,  as  in  the  pulley  or 
rope  machine. 

3.  The  principle  of  oblique  action,  applicable  to  all  the  other 
machines,  the  force  being  employed  to  balance  or  overcome  one 
component  only  of  the  resistance. 

The  force  which  ordinarily  puts  a  machine  in  motion  is  called 
the  power  ;  the  force  which  resists  the  power,  and  is  balanced  or 
overcome  by  it,  is  called  the  resistance,  or  weight. 

A  compound  machine  is  one  in  which  two  or  more  simple  ma- 
chines are  so  connected  that  the  weight  of  the  first  constitutes  the 
power  of  the  second,  the  weight  of  the  second  the  power  of  the 
third.  &c. 


66 


MECHANICS. 


FIG.  62. 


I.  THE  LEVER. 

104.  The  Three  Orders  of  Straight  Lever. —The  lever 

is  a  bar  of  any  form,  free  to  turn  on  a 

fixed  point,  which  is  called  the  ful-     ^ 

crum.     In  the  first  order  of  lever,  the         r 

fulcrum  is  between  the  power  and     •psi 

weight  (Fig.  62) ;  in  the  second,  the 

weight  is  between  the  power  and  ful-  w 

crum  (Fig.  63)  ;  in  the  third,  the  power  is  between  the  weight 

and  fulcrum  (Fig.  64). 


1 


FIG.  63. 


FIG.  64. 


105.  Equal  Moments  in  Relation  to  the  Fulcrum. — 

According  to  the  principle  of  moments,  we  find  for  each  order 
of  the  lever,  P  x  A  G  =  W  x  B  0  ;  that  is, 

The  power  and  weight  have  equal  moments  in  relation  to  the 
fulcrum. 

The  moment  of  either  force  is  the  measure  of  its  efficiency  to 
turn  the  lever  ;  for,  since  the  lever  is  in  equilibrium,  the  efficiency 
of  the  power  to  turn  it  in  one  direction  must  equal  the  efficiency 
of  the  weight  to  turn  it  in  the  opposite  direction.  We  may 
therefore  use  P  x  A  C  to  represent  the  former,  and  W  x  B  C, 
the  latter. 

If  several  forces,  as  in  Fig.  65,  are  in  equilibrium,  some  tending 

FIG,  65. 


to  turn  the  bar  in  one  direction,  and  others  in  the  opposite,  then 
A  and  B  must  have  the  same  efficiency  to  produce  one  motion  as 
C  and  D  have  to  produce  the  opposite ;  that  is,  Ax  A  G-\-BxB  G 
=  CxCG  +  DxDG',  or, 

The  sum  of  the  moments  of  A  and  B  equals  the  sum  of  the  mo- 
ments of  C  and  D. 


ACTING    DISTANCE.  67 

In  order  to  allow  for  the  influence  of  the  weight  of  the  lever 
itself,  consider  it  to  be  collected  at  its  centre  of  gravity,  and  add 
its  moment  to  that  of  the  power  or  weight,  according  as  it  aids 
the  one  or  the  other.  In  Fig.  62,  let  the  weight  of  the  lever  =  w, 
and  the  distance  of  its  centre  from  C  on  the  side  of  P  =.  m  ;  then 
P  x  A  C  +  m  w  —  W  x  B  C.  In  the  2d  and  3d  orders,  the  mo- 
ment of  the  lever  necessarily  aids  the  weight ;  and  hence,  in  each 
case,  P  x  A  C  —  W  x  B  C  +  m  w. 

If  a  weight  hangs  on  a  bar  between  two  supports,  as  in  Fig.  66, 

it  may  be  regarded  as  a  lever  of  the 

•.-,      nf> 

2d  order,  the  reaction  of  either  sup- 
port being  considered  as  a  power. 
Let  F  denote  the  reaction  at  A>  and 
F'  at  C ';  then  by  the  theorems  of 
parallel  forces,  we  have  the  pressures 
at  A  and  C  inversely  as  their  dis- 
tances  from  B,  and  W  =  F  +  F'. 

Eesnming  the  equation  P  x  A  C  -=.W  x  B  C,  we  derive  the 
proportion  P  :  W  ::  B  C  :  A  C ';  hence,  in  each  order  of  the 
straight  lever,  when  the  forces  act  in  parallel  lines,  The  power 
and  iveight  are  inversely  as  the  lengths  of  the  arms  on  which 
they  act. 

106.  The  Acting  Distance. — In  the  three  orders,  as  above 
described,  the  equilibrium  is  not  destroyed  by  inclining  the  lever 
to  any  angle  Avhatever  with  the  horizon,  provided  the  lever  is  sym- 
metrical with  respect  to  its  longer  axis  and  the  centre  of  motion 
C  is  on  this  axis  and  not  above  or  below  it,  and  provided  the  direc- 
tions of  the  forces  remain  vertical.  For  by  the  principle  of 
parallel  forces  any  straight  line  intersecting  the  lines  of  the  forces 
is  divided  by  the  line  of  the  resultant 
into  parts  which  are  inversely  as  the 
forces  ;  therefore  (Fig.  Ql)bC:aC::  AM, 
P  :  W.  Hence,  the  resultant  of  P 
and  W  remains  at  (7,  in  every  position 
of  the  lever.  By  similar  triangles, 
bd  aC  ::  CN  :  CM-,  .-.  P:  W:: 
ON:  CM-,  /.  PxCM=WxCN. 
The  lines  CM  and  C N,  which  are 
drawn  from  the  fulcrum  perpendic- 
ular to  the  lines  in  which  the  forces  act,  are  called  the  acting  dis- 
tances or  the  lever  arms  of  the  power  and  weight,  respectively. 
And  as  they  may  be  employed  in  levers  of  irregular  form,  the 
moments  of  power  and  weight  are  usually  measured  by  the  pro- 
ducts, P  x  CM  and  W  x  ON',  therefore,  the  power  multiplied 


68  MECHANICS. 

ly  its  acting  distance  equals  the  weight  multiplied  "by  its  acting 
distance  ;  or,  more  briefly,  the  moment  of  the  power  equals  the 
moment  of  the  weight,  as  in  Art.  105.  In  Figs.  62,  63,  and  64, 
the  acting  distances  are  in  each  case  identical  with  the  arms  of 
the  lever. 

107.  Lever  not  Straight,  and  Forces  not  Parallel.— 

Let  A  OB  (Fig.  68)  be  a  lever  of  any  form,  and  let  it  be  in  equi- 

librium by  the  forces  P  and  P'9 

acting  in  any  oblique  directions 

in  the  same  plane.     Produce  P  A 

and   P'  B  till  they  meet  in  D  ; 

then,  if  the  fulcrum  is  at  (7,  the 

resultant  must  be  in  the  direction 

D  C\   otherwise   the  reaction  of 

the  fulcrum  cannot  keep  the  sys- 

tem   in    equilibrium    (Art.    43). 

Therefore  (Art  44), 

P  :  P'  :  :  sin  B  D  C  :  sin  A  D  C. 

Draw  CM  perpendicular  to  A  D,  and  ON  to  B  D,  and  they 
are  the  sines  of  A  D  C  and  B  D  C,  to  the  same  radius  D  C. 

.-.  P:P'  ::  ON:  CM-,  and  P  x  C  M  =  P'  x  C  N. 

The  lines  C  M  and  C  JVare  the  acting  distances  of  P  and  P'  ; 
therefore  the  law  of  the  lever  in  all  cases  is  the  same,  namely  : 

TJie  moment  of  the  power  equals  the  moment  of  the  weight. 

When  the  forces  act  obliquely,  the  pressure  on  the  fulcrum  is 
less  than  the  sum  of  the  forces  ;  for,  if  C  E  is  parallel  to  B  D, 
then  D  E,  E  C,  and  C  D,  represent  the  three  forces  which  are  in 
equilibrium.  But  C  D  is  less  than  the  sum  of  D  E  and  E  C. 

108.  The  Compound   Lever.  —  When  a  lever  acts  on  a 
second,  that  on  a  third,  &c.,  the  machine  is  called  a  compound 
lever.     The  law  of  equilibrium  is  — 

The  continued  product  of  the  power  and  acting  distances  on  the 
side  of  the  power  is  equal  to  the  continued  product  of  the  weight 
and  acting  distances  on  the  side  of  the  weight. 

Let  the  force  exerted  by  A  B  on  B  D  (Fig.  69)  be  called  x.  and 
that  of  B  D  on  D  E  be  called  y  ;  then 


x   x  B  F  =  y  x  DF\ 
y  x  DG  =W  x  QR 


THE    BALANCE.  69 

Multiply  these  equations  together  and  omit  common  factors,  and 
we  have 

PxACxBFxDG=WxBGxDFxGK 
FIG.  69. 

A  C    B T -?  G 


If  the  levers  were  of  irregular  forms,  the  acting  distances  might 
not  be  identical  with  the  arms,  as  they  are  in  the  figure. 

109.  The  Balance. — This  is  a  common  and  valuable  instru- 
ment for  weighing.  It  is  a  straight  lever  with  equal  arms,  having 
scale-pans,  either  suspended  at  the  ends,  or  standing  upon  them, 
one  to  contain  the  poises,  and  the  other  the  substance  to  be 
weighed.  For  scientific  purposes,  particularly  for  chemical  analy- 
sis, great  care  is  bestowed  on  the  construction  of  the  balance. 

The  arms  of  the  balance,  measured  from  the  fulcrum  to  the 
points  of  suspension,  must  be  precisely  equal. 

The  knife-edges  forming  the  fulcrum,  and  the  points  of  sus- 
pension, are  made  of  hardened  steel,  and  arranged  exactly  in  a 
straight  line. 

The  centre  of  gravity  of  the  beam  is  beloiv  the  fulcrum,  so  that 
there  may  be  a  stable  equilibrium  ;  and  yet  below  it  by  an  exceed- 
ingly small  distance,  in  order  that  the  balance  may  be  very 
sensitive. 

To  preserve  the  edge  of  the  fulcrum  from  injury,  the  beam  is 
raised  by  supports  called  Y's,  when  not  in  use. 

A  long  index  at  right  angles  to  the  beam,  points  to  zero  on  a 
scale  when  the  beam  is  horizontal. 

To  protect  the  instrument  from  dust  and  moisture  at  all  times, 
and  from  air-currents  while  weighing,  the  balance  is  in  a  glass 
case,  whose  front  can  be  raised  or  lowered  at  pleasure. 

A  balance  for  chemical  analysis  is  shown  in  Fig.  70.  By  turn- 
ing the  knob  0,  the  beam  can  be  raised  on  the  Y's  A  A  from  the 
surface  on  which  the  fulcrum  K  rests.  The  screw  C  raises  and 
lowers  the  fulcrum  in  relation  to  the  centre  of  gravity  of  the  beam, 
in  order  to  increase  or  diminish  the  sensitiveness  of  the  instru- 
ment. In  the  most  carefully  made  balances,  the  index  will  make 


70 


MECHANICS. 


a  perceptible  change,  by  adding  to  the  scale  one  millionth  of  the 
poise. 

FIG.  70. 


For  commercial  purposes,  it  is  convenient  to  have  the  scale- 
pans  above  the  beam.  This  is  done  by  the  use  of  additional  bars, 
which  with  the  beam  form  parallelograms,  whose  upright  sides 
are  rods,  projecting  upward  and  supporting  the  scales.  Such  con- 
trivances necessarily  increase  friction  ;  but  balances  so  constructed 
are  sufficiently  sensitive  for  ordinary  weighing. 

110.  The  Steelyard. — This  is  a  weighing  instrument,  hav- 
ing a  graduated  arm,  along  which  a  poise  may  be  moved,  in  order 
to  balance  various  weights  on  the  short  arm.  While  the  moment 
of  the  article  weighed  is  changed  by  increasing  or  diminishing  its 
quantity,  that  of  the  poise  is  changed  by  altering  its  acting  dis- 
tance. Since  P  x  AC—  W  x  B  C  (Fig.  71),  and  P  is  constant, 

FIG.  71. 


[nrinfiiimpiqiiiinni.ftmimimimi.ni 


and  also  the  distance  B  C  constant,  A  C  QC  W',  hence,  if  W  is 
successively  1  lb.,  2  Ibs.,  3  Ibs.,  &c.,  the  distances  of  the  notches. 


PLATFORM    SCALES. 


71 


«,  b9  c,  &c.,  are  as  1,  2,  3,  &c.  ;  in  other  words,  the  bar  CD  is 
divided  into  equal  parts.  In  this  case,  the  graduation  begins 
from  the  fulcrum  C  as  the  zero  point. 

But  suppose,  what  is  often  true,  that  the  centre  of  gravity  of 
the  steelyard  is  on  the  long  arm,  and  that  P  placed  at  E  would 
balance  it  ;  then  the  moment  of  the  instrument  itself  is  on  the 
side  C  D,  and  equals  P  x  C  E.     Hence,  the  equation  becomes 
P  x  A  C+  P  x  CE  =  W  x  B  6';  or 


.:  JFoc  A  E\  and  the  graduation  must  be  considered  as  com- 
mencing at  E  for  the  zero  point.  Such  a  steelyard  cannot  weigh 
below  a  certain  limit,  corresponding  to  the  first  notch  a. 

To  find  the  length  of  the  divisions  on  the  bar,  divide  A  E,  the 
distance  of  the  poise  from  the  zero  point,  by  W,  the  number  of 
units  balanced  by  P,  when  at  that  distance. 

The  steelyard  often  has  two  fulcrums,  one  for  less  and  the  other 
for  greater  weights. 

111.  Platform  Scales.  —  This  name  is  given  to  machines 
arranged  for  weighing  heavy  and  bulky  articles  of  merchandise. 
The  largest,  for  cattle,  loaded  wagons,  &c.,  are  constructed  with 
the  platform  at  the  surface  of  the  ground.  In  order  that  the  plat- 
form may  stand  firmly  beneath  its  load,  it  rests  by  four  feet  on  as 
many  levers  of  the  second  order,  whose  arms  have  equal  ratios. 
A  F,  B  F,  C  G,  D  G  (Fig.  72),  are  four  such  levers,  resting  on  the 

FIG.  72. 


fulcrums,  A,  B,  (7,  D,  while  the  other  ends  meet  on  the  knife- 
edge,  F  G,  of  another  lever,  L  M.  This  fifth  lever  has  its  fulcrum 
at  Z,  and  its  outer  extremity  is  attached  by  a  vertical  rod,  M  N, 
to  a  steelyard,  whose  fulcrum  is  E,  and  poise  P.  The  five  levers 
are  arranged  in  a  square  cavity  just  below  the  surface  of  the 
ground.  The  dotted  line  shows  the  outline  of  the  cavity.  On  the 
bearing  points  of  the  four  levers,  H,  1,  J,  J£,  rest  the  feet  of  the 
platform  (not  represented),  which  is  firmly  built  of  plank,  and  just 


72  MECHANICS. 

fits  into  the  top  of  the  cavity  without  touching  the  sides.  The 
machine  is  a  compound  lever  of  three  parts  ;  for  the  four  levers 
act  as  one  at  F  G,  and  are  used  to  give  steadiness  to  the  platform 
which  rests  upon  them. 

A  construction  quite  similar  to  the  above  is  made  of  portable 
size,  and  used  in  all  mercantile  establishments  for  weighing  heavy 
goods. 

112.  Questions  on  the  Lever.— 1.  A  B  (Fig.  73)  is  a 
uniform  bar,  2  feet  long,  and 

Tfyo       net 

weighs  4  oz. ;  where  must  the  f  ul-  B  c 

crum  be  put,  that  the  bar  may  be  A 

balanced  by  P,  weighing  5  Ibs.  ? 

Am.  4  of  an  inch  from  A.  I 

2.  A  lever  of  the  second  order  ]  9 
is  25  feet  long  ;  at  what  distance  from  the  fulcrum  must  a  weight 
of  125  pounds  be  placed,  so  that  it  may  be  supported  by  a  power 
able  to  sustain  60  pounds,  acting  at  the  extremity  of  the  lever. 

Ans.  12  feet. 

3.  A  and  B  are  of  the  same  height,  and  sustain  upon  their 
shoulders  a  weight  of  150  pounds,  placed  on  a  pole  9£  feet  long  ; 
the  weight  is  placed  6|  feet  from  A  ;  what  is  the  weight  sustained 
by  each  person  ? 

Ans.  A  sustains  42|  Ibs.,  and  B  sustains  107^  Ibs. 

4.  A  bent  lever,  A  C  B  (Fig.  74),  has  the  arm  A  C  =  3  feet, 
C  B  =  8  feet,   P  =  5  Ibs.,  and  the 

angle  A  C  B  =  140° ;  what  weight, 
W,  must  be  attached  at  £,  in  order  to 
keep  A  C  horizontal  ? 

Am.  2.4476  Ibs. 

5.  A  cylindrical  straight  lever  is 
14  feet  long,  and  weighs  6  Ibs.  5  oz. ; 
its  longer  arm  is  9,  and  its  shorter  5 
feet ;  at  the  extremity  of  its  shorter 
arm  a  weight  of  15  Ibs.  2  oz.  is  sus- 
pended ;  what  weight  must  be  placed 
at  the  extremity  of  the  longer  arm  to 

keep  it  in  equilibrium  ?  Ans. 

6.  A  uniform  bar,  12  feet  long,  weighs  7  Ibs. ;  a  weight  of  10 
Ibs.  hangs  on  one  end,  and  2  feet  from  it  is  applied  an  upward 
force  of  25  Ibs.,  where  must  the  fulcrum  be  put  to  produce  equi- 
librium? Ans.  1  foot  from  the  10  Ibs. 

7.  The  lengths  of  the  arms  of  a  balance  are  a  and  b.     When  p 
ounces  are  hung  on  «,  they  balance  a  certain  body;  but  it  re- 


WHEEL    AND    AXLE. 


73 


quires  q  ounces  to  balance  the  same  body,  when  placed  in  the 
other  scale.  What  is  the  true  weight  of  the  body  ?  According  to 
the  first  weighing,  a  p  =  b  x ;  according  to  the  second,  b  q  =  a  x. 
.:  a  b  p  q  :  =  a  b  x*,  and  x  =  V  p  q-  Hence,  the  true  weight  is 
a  geometrical  mean  between  the  apparent  weights. 

8.  On  one  arm  of  a  false  balance  a  body  weighs  11  Ibs.,  on  the 
other,  17  Ibs.  3  oz.;  what  is  the  true  weight  ? 

Am.   13  Ibs.  12  oz. 

9.  Four  weights  of  1,  3,  5,  7  Ibs.,  respectively,  are  suspended 
from  points  of  a  straight  lever,  eight  inches  apart ;  how  far  from 
the  point  of  suspension  of  the  first  weight  must  the  fulcrum  be 
placed,  that  the  weights  may  be  in  equilibrium  ? 

Ans.  17  inches. 

10.  Two  weights  keep  a  horizontal  lever  at  rest,  the  pressure 
on  the  fulcrum  being  10  Ibs.,  the  difference  of  the  weights  4  Ibs., 
and  the   difference  of  the  lever  arms  9  inches  ;   what  are  the 
weights  and  their  lever  arms  ? 

Ans.  Weights,  7  Ibs.  and  3  Ibs.;  arms,  6|  in.  and  15f  in. 

• 

II.  THE  WHEEL  AND  AXLE. 

113.  Description  and  Law  of  the  Machine.— The  wheel 
and  axle  consists  of  a  cylinder  and  a  wheel,  firmly  united,  and  free 
to  revolve  on  a  common  axis.  The  power  acts  at  the  circumfer- 
ence of  the  wheel  in  the  direction  of  a  tangent,  and  the  weight  in 
the  same  manner,  at  the  circumference  of  the  cylinder  or  axle ;  so 
that  the  acting  distances  are  the  radii  at  the  two  points  of  contact. 
As  the  system  revolves,  the  radii  successively  take  the  place  of 
acting  distances,  without  altering  at  all  the  relation  of  the  forces 
to  each  other.  The  wheel  and  axle  is  therefore  a  kind  of  endless 
lever. 

Let  W  (Fig.  75)  be  the  weight  suspended  from  the  axle,  tend- 
ing to  revolve  it  on  the  line  L  M ; 
and  P,  the  power  acting  on  the 
wheel,  tending  to  revolve  the  sys- 
tem in  the  opposite  direction.  It 
is  plain  that  the  acting  distances 
are  the  radius  of  the  axle,  and  A  C 
the  radius  of  the  wheel.  In  case  of 
equilibrium,  the  moment  of  W 
equals  the  moment  of  P.  Calling 
the  radius  of  the  axle  r,  and  the 
radius  of  the  wheel  R,  then  W  x  r 
=  P  x  R ;  or 

P  :W  r.riR. 


FIG.  75. 


74 


MECHANICS. 


FIG.  7( 


If,  instead  of  the  weight  P,  suspended  on  the  wheel,  the  rope 
be  drawn  by  any  force  in  the  direction  P'  or  P",  it  is  still  tangent 
to  the  circumference,  and  therefore  its  acting  distance,  CD  or  C B> 
the  same  as  before.  In  general,  the  law  of  equilibrium  for  this 
machine  is, 

The  moment  of  the  Power  is  equal  to  the  moment  of  the 
Weight. 

If  the  rope  on  the  wheel,  being  fastened  at  A  (Fig.  76)  i» 
drawn  by  the  side  of  the  wheel,  as  A  P', 
the  acting  distance  of  the  power  is  dimin- 
ished from  CA  to  C  E,  and  therefore  its 
efficiency  is  diminished  in  the  same  ratio. 
Were  the  rope  drawn  away  from  the 
wheel,  as  A  P",  making  an  equal  angle 
on  the  other  side  of  A  P,  the  same  effect 
is  produced,  the  acting  distance  now  be- 
coming C  F. 

Tfie  radius  of  the  wheel  and  the  radius 
of  the  axle  should  each  be  reckoned  from 
the  axis  of  rotation  to  the  centre  of  the 
rope  ;  that  is,  half  of  the  thickness  of  the  rope  should  be  added  to 
the  radius  of  the  circle  on  which  it  is  coiled.  Calling  t  the  half 
thickness  of  the  rope  on  the  axle,  and  t'  that  of  the  rope  on  the 
wheel,  the  equation  of  equilibrium  is 

P  x  (R  +  t1)  =  W  x  (r  +  t). 

In  considering  the  wheel  and  axle  no  account  has  been  taken 
of  the  stiffness  of  the  rope,  which  acts  as  a  constant  resistance, 
opposing  motion  in  winding  upon  a  drum  or  wheel,  and  also  in 
unwinding. 

114.  Differential  Pulley. — A  modification  of  the  wheel  and 
axle,  called  a  differential  pulley,  is  of  great  use  in  raising  very 
heavy  weights  through  short  distances. 

The  pulley  consists  of  a  solid  wheel  A  (Fig.  77),  one  half  of 
which,  #,  is  of  less  diameter  than  the  other  half,  a, 
suspended  in  a  block  in  the  usual  manner. 

A  continuous  chain  is  used,  which  we  may  trace 
from  the  point  A  (Fig.  78),  upward,  over  the  larger 
of  the  two  circumferences,  then  downward  through 
B  to  the  movable  pulley  D,  thence  upward  through 
(7  around  the  smaller  circumference  of  the  wheel, 
thence  down  through  E  and  back  to  the  point  of 
beginning  at  A. 
Call  the  radii  R  and  r  as  indicated  in  the  figure,  and  suppose 


FIG.  77. 


1 


COMPOUND   WHEEL  AND  AXLE. 


75 


E 


a  downward  force  P  to  be   applied  to  the  chain 
P  x  R-\-^Wxr  =  ^Wx  R,  in  which  equa- 
tion no  account  is  taken  of  the  weight  of  the  chain. 
Transposing,  we  obtain, 

P  x  R  =  i  W  (R-r)  or 

P:$  W  ::R-r:R. 

Now  R — r  may  be  made  as  small  as  we  please, 
and  hence  the  power  also  may  be  made  small  as  com- 
pared with  the  weight.  The  weight  of  the  chain 
and  the  friction  act  as  resistances  to  motion,  and  are 
sufficient  to  prevent  the  downward  run  of  D  after 
the  hand  is  removed  from  A,  even  when  W  is  very 
great.  This  pulley  may  be  found  in  any  large  V 
foundry,  or  machine  shop. 

115.  The  Compound  Wheel  and  Axle.— 

When  a  train  of  wheels,  like  that  in  Fig.  79,  is  put 
in  motion,  those  which  communicate  motion  by  the 
circumference  are  called  driving  wheels,  as  A  and  (7; 
those  which  receive  motion  by  the  circumference 
are  called  driven  wheels.  And  the  law  of  equili- 
brium is, 

The  continued  product  of  the  poiver  and  radii  of 
the  driven  wheels  is  equal  to  the  continued  product  of  the  weight 
and  radii  of  the  driving  ivheels. 

The  crank  P  Q  is  to  be  reckoned  among  driven  wheels ;  the 
axle  E  among  driving  wheels. 

Let  the  radius  of  B  be  called  R ;  of 
D,  R'',  of  A,  r\   of  C,  r' ;  of  E,  r". 
Call  the  force  exerted  by  A  on  B,  #; 
that  of  C  on  Z>,  y.     Then 
P  x  P  Q=  r  x  a?; 
x  x  R  =  r'  x  y  ; 
y  x  R'  =  r"  x   W. 
Multiply  and  omit  common  factors,  and 
we  have 
PxP  QxRxR'=Wxr"xr'xr. 

If  the  driving  wheels  are  equal  to  each  other,  and  also  the 
driven  wheels,  and  the  number  of  each  is  n,  then 
P  x  Rn  —  W  r\ 

116.  Direction  and  Rate  of   Revolution.— When   two 

wheels  are  geared  together  by  teeth,  they  necessarily  revolve  in 
contrary  directions.  Hence,  in  a  train  of  wheels,  the  alternate 
axles  revolve  the  same  way. 

The  circumferences  of  two  wheels  which  are  in  gear  move  with 


FIG.  79. 


76  MECHANICS. 

tiie  same  velocity  ;  hence  the  number  of  revolutions  will  be  recip- 
rocally as  the  radii  of  the  wheels. 

Since  teeth  which  gear  together  are  of  the  same  size,  the  rela- 
tive number  of  teeth  is  a  measure  of  the  relative  circumferences, 
and  therefore  of  the  relative  radii  of  the  wheels.  If  the  wheel  A 
(Fig.  79)  has  20  teeth,  and  B  has  40,  and  again  if  G  has  15,  and 
D  45,  then  for  every  revolution  of  B,  A  revolves  twice,  and  for 
every  revolution  of  D,  C  revolves  three  times.  Therefore,  six 
turns  of  the  crank  are  necessary  to  give  one  revolution  to  the 
axle  E. 

By  cutting  the  teeth  of  wheels  on  a  conical  instead  of  a  cylin- 
drical surface,  the  axles  may  be  placed  at  any  angle  with  each 
other,  as  represented  in  Fig.  80. 

Whether  axles  are  parallel  or  not,  bands  in- 
stead of  teeth  may  be  used  for  transmitting 
rotary  motion.  But  as  bands  are  liable  to  slip 
more  or  less,  they  cannot  be  employed  in  cases 
requiring  exact  relations  of  velocity. 

117.  Questions  on  the  Wheel  and 
Axle.— 

1.  A  power  of  12  Ibs.  balances  a  weight  of 
100  Ibs.  by  a  wheel  and  axle  ;  the  radius  of  the 
axle  is  6  inches  ;  what  is  the  diameter  of  the  wheel  ? 

Ans.  8  ft.  4  in. 

2.  PF=500  Ibs.;  R  =  4  ft.;  r  =  8  in.;  the  weight  hangs  by  a 
rope  1  inch  thick,  but  the  power  acts  at  the  circumference  of  the 
wheel  without  a  rope  ;  what  power  will  sustain  the  weight  ? 

Ans.  88.54  Ibs. 

3.  In  Fig.  79,  A  and  C  have  each  15  teeth,  B  and  D  each  40 
teeth  ;  the  radius  of  the  axle  E  is  4  inches  ;  the  rope  on  it  1  inch 
in  diameter  ;  and  the  radius  of  the  crank  P  Q  is  18  inches  ;  what 
is  the  ratio  of  power  to  weight  in  equilibrium?  Ans.  1  : 


118.  The  Pulley  Described.  —  The  pulley  consists  of  one 
or  more  wheels  or  rollers,  with  a  rope  passing  over  the  edge  in 
which  a  groove  is  sunk  to  keep  the  rope  in  place.  The  axis  of 
the  roller  is  in  a  block,  which  is  sometimes  fixed,  and  sometimes 
rises  and  falls  with  the  weight  ;  and  the  pulley  is  accordingly 
called  a  fixed  pulley  or  a  movable  pulley.  The  principle  which 
explains  the  relation  of  power  and  weight  in  every  form  of 
pulley  is  this  : 

WJiatever  strain  or  tension  is  applied  to  one  end  of  a  cord,  is 
transmitted  through  its  ivhole  length,  if  it  does  not  branch,  however 
much  its  direction  is  changed. 


THE    FIXED    PULLEY. 


77 


FIG.  81. 


FIG.  82. 


mw 


In  the  pulley,  the  sustaining  portions  of  the  rope  are  assumed 
to  be  parallel  to  each  other. 

119.  The  Fixed  Pulley.—  In 
the  fixed  pulley,  A  (Fig.  81),  the 
force  P,  produces  a  tension  in  the 
string,  which  is  transmitted  through 
its  whole  length,  and  which  can  be 
balanced  only  when    W  equals  P. 
Hence,  in    the    fixed  pulley,   the 
poiver  and  weight  are  equal.     This 
machine  is  useful  for  changing  the 
direction  in  which  the  force  is  ap- 
plied to  the  weight:    and  if  the 
power  only  acts  in  the  plane  of  the 

groove  of  the  wheel,  it  is  immaterial  what  is  its  direction,  horizon- 
tal,  vertical,  or  oblique. 

120.  The  Movable  Pulley.—  In  Fig.  82,  the       FIG.  83. 
tension  produced  by  P,  is  transmitted  from  A  down 

to  the  wheel  E,  and  thence  up  to  D  ;  therefore  W  is 
sustained  by  two  portions  of  the  rope,  each  of  which 
exerts  a  force  equal  to  P. 

.'.  W  =  2  P-,  or  P  :  W::  1  :  2. 

The  same  reasoning  applies,  where  the  rope  passes 
between  the  upper  and  lower  blocks  any  number  of 
times,  as  in  Fig.  83.  The  force  causes  a  tension  in  the 
rope,  which  is  transmitted  to  every  portion  of  it.  If  n 
is  the  number  of  portions  which  sustain  the  lower 
block,  then  W  is  upheld  by  n  P  ;  and  if  there  is  equi- 
librium, P  :  W  :  :  1  :  n.  In  the  figure,  the  weight 
equals  six  times  the  power.  The  law  of  equilibrium, 
therefore,  for  the  movable  pulley  with  one  rope,  is  this, 

TJie  power  is  to  the  iveight  as  one  to  the  number  of 
the  sustaining  portions  of  the  rope. 

121.  The  Compound  Pulley.—  Wherever  a  system  of  pul- 
leys has  separate  ropes  the  machine  is  to  be  regarded  as  com- 
pound, and  its  efficiency  is  calculated  accordingly.     Figures  84 
and  85  are  examples.     In  Fig.  84  call  the  weight  sustained  by  F9  x> 
and  that  sustained  by  D,  y.     Then  (Art.  120), 

P  :    x::  I  :2 


y  :  W  :  :  1  :  2. 
/.  P:  JF::1:23::1:8. 

And  if  n  is  the  number  of  ropes,  P  :  W  :  :  1  : 


2*. 


78 


MECHANICS. 


In  Fig.  85  the  tension  P  is  transmitted  over  A  directly  to  the 
weight  at  G ;  the  wheel  A  is  loaded,  therefore,  with  2  P,  and  a 
tension  of  2  P  comes 
upon  the  second  rope, 
which  is  transmitted 
over  B  to  the  weight  at 
F.  In  like  manner,  a 
tension  of  4  P  is  trans- 
mitted over  0  to  E. 
The  sum  of  all  these 
being  applied  to  the 
weight,  it  must  there- 
fore be  equal  to  that 
sum  in  case  of  equili- 
brium. Therefore,  P : 


\v 


w 

Now  the  sum  of  this 

geometrical  series  to  n  terms  is  2B  —  1  , 
/.  P  :  Wr.  I  :  2"  —  1.  This  combination  is 
therefore  a  little  less  efficient  than  the  pre- 
ceding. 

Since  the  several  ropes  have  different  ten- 
sions, the  weight  cannot  be  balanced  upon 
them,   unless  those  of  greatest  tension  are 
nearest  the  line  of  direction  of   the  body. 
For  example,  if  the  rope  F  is  directed  toward 
the  centre  of  gravity  of  the  weight,  the  rope  G  should  be  attached 
four  times  as  far  from  it  as  the  rope  E,  in  order  to  prevent  the 
weight  from  tipping. 

The  pulley  owes  its  efficiency  as  a  machine  to  the  fact,  that  the 
tension  produced  by  the  power  is  applied  repeatedly  to  the  weight. 
The  only  use  of  the  wheels  is  to  diminish  friction.  Were  it  not 
for  friction,  the  rope  might  pass  round  fixed  pins  in  the  blocks, 
and  the  ratio  of  power  to  weight  would  still  be  in  every  case  the 
same  as  has  been  shown. 


IV."  THE  ROPE  MACHINE. 

122.  Definition  and  Law  of  this  Machine.— 

TJie  rope  machine  is  one  in  which  the  power  and  weight  are  in 
equilibrium  by  the  tension  of  one  or  more  ropes. 

According  to  this  definition  the  pulley  is  included.  It  is  that 
particular  form  of  the  rope  machine  in  which  the  sustaining  parts 
of  the  ropes  are  parallel ;  and  it  is  treated  as  a  separate  machine. 


THE    ROPE    MACHINE. 


79 


because  its  theory  is  very  simple,  and  because  it  is  used  far  more 
extensively  than  any  other  forms. 

If  the  two  portions  of  rope  which  sustain  the  weight  are 
inclined,  as  in  Fig.  86,  then  W  is  no  longer  equal  to  the  sum  of 
their  tensions,  as  it  is  in  the 
pulley,  but  is  always  less  than  FlG-  86- 

that,  according  to  the  follow-  f  fi      , 

ing  law  : 

The  power  is  to  the  iveight 
as  the  sine  of  -J-  the  angle  is  to 
the  sine  of  the  whole  angle  be- 
tween the  parts  of  the  rope. 

Put  A  E  B  —  2  a  ;  then 
FED—  a,  and  since  sin  B  E  W 
=  sin  B  ED  =  sin  a,  we  shall 
have  (Art.  44)  P  :  W : :  sin  a  : 
sin  2  a. 

If  in  Fig.  87,  the  end  of  the 
cord,  instead  of  being  attached 

to  the  beam,  is  carried  over  another  fixed  pulley,  and  a  weight 
equal  to  P  is  hung  upon  it,  the  equilibrium  will  be  preserved, 
because  all  parts  of  the  rope  have  a  tension  equal  to  P\  therefore, 
as  before, 

P  :  W  ::  sin  a  :  sin  2  a. 

123.  Change  in  the  Ratio  of  Power  and  Weight. — 
If  P  is  given,  all  the  possible  values  of  W  are  included  between 
W  =  0,  and  W  =  2  P. 

When  the  rope  is  straight  from  A  to  B  (Fig.  87),  so  that 
C  D  —  0,  then,    by  the   above  proportion,    W  =0.      As  IT  is 
increased  from  zero,  the  point 
C  descends  ;  and  when  D  C  = .  Fia  87- 

\B  C>  then,  by  the  proportion, 
W  =  P.  In  jthat  case  DC  B 
=  60°,  and  the  angles  A  C  B, 


ACW,  and  B  O  W  are  equal 
(each  being  120°),  as  they 
should  be,  because  each  of  the 
equal  forces,  P9  P,  and  W,  is 
as  the  sine  of  the  angle  between  the  directions  of  the  other  two. 

But  when  W  has  increased  to  2  P,  it  descends  to  an  infinite 
distance  ;  for  then,  by  the  proportion,  CD  =  B  C,  that  is,  the  side 
of  a  right-angled  triangle  is  equal  to  the  hypothenuse.  Thus,  the 
extreme  values  of  W  are  0  and  2  P. 

It  appears   from  the  foregoing,  that  a  perfectly  flexible  rope 


80 


MECHANICS. 


having  iveight  cannot  be  drawn  into  a  straight  horizontal  line,  by 
any  force  however  great ;  for  G  cannot  coincide  with  Z>,  except 
when  W  =  0. 

124.  The  Branching  Rope. — When  (7,  where  the  weight  is 
suspended,  is  a.  fixed  point  of  the  rope, 

we  have  a  branching  rope,  and  the 
principle  of  transmitted  tension  does 
not  apply  beyond  the  point  of  division. 
Let  P,  P'  and  W  (Fig.  88),  be 
given,  and  0  a  fixed  point  of  the  rope. 
Produce  W  C9  and  let  A  E9  drawn 
parallel  to  OS,  intersect  it  in  E.  The 
sides  (A  A  G  E  are  proportional  to  the 
given  forces ;  therefore  its  angles  can 
be  found,  and  the  inclinations  of  A  G 
and  B  G  to  the  vertical  G  W  are  known. 

V.  THE  INCLINED  PLANE. 

125.  Relation  of  Power,  Weight,  and  Pressure  on 
the  Plane. — The  mechanical  efficiency  of  the  inclined  plane  is 
explained  on  the  principle  of  oblique  action;  that  is,  it  enables  us 
to  apply  the  power  to  balance  or  overcome  only  one  component  of 
the  weight,  instead  of  the  whole.     Let  the  weight  of  the  body  G, 
lying  on  the  inclined  plane  A  G  (Fig.  89),  be  represented  by  W  > 
and  resolve  it  into  F  parallel,  and  N  perpendicular  to  the  plane. 
N  represents  the  perpendicular  pressure,  and  is  equal  to  the 
reaction  of  the  plane ;  F  is  the  force  by  which  the  body  tends  ta 
move  down  the  plane. 

Let  a  =  the  angle  (7,  the  inclination  of  the  plane ;  therefore 
W  D  N  =  a.    Then  F  —  W .  sin  a  ;  and  N  =  W .  cos  a. 


FIG.  89. 


FIG.  90. 


Now  suppose  a  force  P  is  applied  at  G  (Fig.  90),  which  keeps 
the  body  at  rest.  Then  the  resultant  of  W  and  P  must  be  N, 
which  is  resisted  by  the  plane  ;  therefore, 

P  :  W  : :  sin  G  N  P,  or  sin  a  :  sin  P  G  N. 


THE    INCLINED    PLANE.  81 

When  the  power  acts  parallel  to  the  plane,  P  G  N  =  90°,  and 
we  have  P  :  W  : :  sin  a  :  sin  90°  : :  A  B  :  A  C.  Hence,  when  the 
power  acts  in  a  line  parallel  to  the  inclined  plane,  which  is  the 
most  common  direction, 

The  power  is  to  the  weight  as  the  height  to  the  length  of  the 
inclined  plane. 

When  the  power  acts  in  a  line  parallel  to  the  base  of  the  in- 
clined plane,  P  G  N  =  90°  —  a,  and  we  have  P  :  W  ::  sin  a  : 
cos  a  : :  A  B  :  B  C.  Hence,  when  the  power  acts  in  a  line  parallel 
to  the  base  of  the  inclined  plane, 

The  power  is  to  the  weight  as  the  height  is  to  the  base  of  the 
inclined  plane. 

126.  Power  most  Efficient  when  Acting  Parallel  to 
the  Plane. — From  the  proportion 

P  :  W  : :  sin  a  :  sin  P  G  N,  we  derive 
P  .  sin  P  G  N 


W  = 


sin  a 


Now  as  P  and  sin  a  are  given,  W  varies  as  sin  P  G  N,  which 
is  the  greatest  possible  when  P  G  N  =  90°  ;  that  is,  when  the 
power  acts  in  a  line  parallel  to  the  plane. 

Whether  the  angle  PON  diminishes  or  increases  from  90°, 
its  sine  diminishes,  and  becomes  zero,  when  P  G  JW  =  0°,  or  180°. 
Therefore  W  =  0,  or  no  weight  can  be  sustained,  when  the  power 
acts  in  the  line  G  N,  perpendicular  to  the  plane,  either  toward 
the  plane  or  from  it. 

127.  Expression  for  Perpendicular  Pressure.  —  From 
the  triangle  P  G  JVwe  obtain 

^:TT::sin  G-  P  N  :  sin  P  G  N, 
or  N\  W::  sin  P  G  W:  sin  P  G  N\ 


sin  P  G  N 
If  the  power  acts  in  a  line  parallel  to  the  inclined  plane, 

P  G  W  =  90°  +  a,  P  G  N  =  90°,  and  N  =  lFffln(.90°  t**  = 

sin  9U 

W  cos  a. 

If  the  power  acts  in  a  line  parallel  to  the  base  of  the  inclined 

plane,  P  G  W  =  90°,  PGN=  90°  —  a,  and  JV=  JF^-^  = 
W  sec  a. 


MECHANICS. 

If  the  power  acts  in  a  line  perpendicular  to  the  inclined  plane, 
G  W  =  a,  P  G  N  =  0°, 


T>r       „,  sm  a 

N  =  W—Q—  =  &  • 


FIG.  91. 


128.  Equilibrium  between  Two  Inclined  Planes. — If  a 

body  rests,  as  represented  in  Fig.  91,  between  two  inclined  planes, 
the  three  forces  which  retain  it  are  its  weight,  and  the  resistances 
of  the  planes.  Draw  H  F  and  L  F 
perpendicular  to  the  planes  through 
the  points  of  contact,  and  G  F  verti- 
cally through  the  centre  of  gravity  of 
the  body.  Since  the  body  is  in  equi- 
librium, these  three  lines  will  pass 
through  the  same  point  (Art.  43).  Let 
that  point  be  F,  and  draw  G  P  paral- 
lel to  L  F,  and  M  K  parallel  to  the 
horizon.  G  P  Fis  similar  to  K  CM. 
Therefore  (since  Pressure  on  A  0  :  Pr. 
on  DC::  P  G  :  F  P), 


Pressure  on  A  C :  Pr.  on  D  C : 


KC:MC, 

sin  M  :  sin  K, 

sin  D  C  E  :  sin  A  C  B. 


That  is,  when  a  body  rests  between  two  planes,  it  exerts  pressures 
on  them  which  are  inversely  as  the  sines  of  their  inclinations  to 
the  horizon. 

If,  therefore,  one  of  the  planes  is  horizontal,  none  of  the 
pressure  can  be  exerted  on  any  other  plane.  It  is  friction  alone 
which  renders  it  possible  for  a  body  on  a  horizontal  surface  to 
lean  against  a  vertical  ivall. 

129.  Bodies  Balanced  on  Two  Planes  by  a  Cord 
passing  over  the  Ridge.— Let  P  and  W  balance  each  other  on 
the  planes  A  D  and  A  C  (Fig.  92),  FlQ  93 

which  have  the  common  height  A  B, 
by  means  of  a  cord  passing  over  the 
fixed  pulley  A.  The  tension  of  the 
cord  is  the  common  power  which  pre- 
vents each  body  from  descending ;  and 
as  the  cord  is  parallel  to  each  plane, 
we  have  (calling  the  tension  t), 

t  :  P  : :  A  B  :  A  D  ; 

and*:  W ::  A  B  :  A  (7; 

.'.P:  W:'.A  D  :A  0- 


THE    SCREW.  ^3 

that  is,  the  weights,  in  case  of  equilibrium,  are  directly  as  the 
lengths  of  the  planes. 

130.  Questions  on  the  Inclined  Plane. — 

1.  If  a  horse  is  able  to  raise  a  weight  of  440  Ibs.  perpendicu- 
larly, what  weight  can  he  raise  on  a  railway  having  a  slope  of  five 
degrees  ?  Ans.  5048.5  Ibs. 

2.  The  grade  of  a  railroad  is  20  feet  in  a  mile  ;  what  power 
must  be  exerted  to  sustain  any  given  weight  upon  it? 

Ans.  1  Ib.  for  every  264  Ibs. 

3.  What  force  is  requisite  to  hold  a  body  on  an  inclined  plane, 
by  pressing  perpendicularly  against  the  plane? 

Ans.  An  infinite  force. 

4.  A  certain  power  was  able  to  sustain  500  tons  on  a  plane  of 
7-J0  ;  but  on  another  plane,  it  could  sustain  only  400  tons  ;  what 
was  the  inclination  of  the  latter  ?  Ans.  9°  23'  25". 

5.  Equilibrium  on  an  inclined  plane  is  produced  when  the 
power,  weight,  and  perpendicular  pressure  are,  respectively,  9,  13, 
and  6  Ibs.;  what  is  the  inclination  of  the  plane,  and  what  angle 
does  the  power  make  with  the  plane? 

Ans.  a  =  37°  21'  26".     Inclination  of  power  to  plane 
=  28°  46'  54". 

6.  A  power  of  10  Ibs.,  acting  parallel  to  the  plane,  supports  a 
certain  weight ;  but  it  requires  a  power  of  12  Ibs.  parallel  to  the 
base  to  support  it.     What  is  the  weight  of  the  body,  and  what  is 
the  inclination  of  the  plane  ? 

Ans.    W  =  18.09  Ibs.    a  =  33°  33'  25". 

7.  To  support  a  weight  of  500  Ibs.  upon  an  inclined  plane  of 
50°  inclination  to  the  horizon,  a  lifting  force  is  applied  whose 
direction  makes  an  angle  of  75°  with  the  horizon.     What  is  the 
magnitude  of  this  force,  and  the  pressure  of  the  weight  against 
the  plane  ?  Ans.  P  =  422.6  Ibs.     N  =  142.8  Ibs. 

VI.    THE  SCREW. 

131.  Reducible  to  the  Inclined  Plane. — The  screw  is  a 
cylinder  having  a  spiral  ridge  or  thread  around  it,  which  cuts  at  a 
constant  oblique  angle  all  the  lines  of  the  surface  parallel  to  the 
axis  of  the  cylinder.     A  hollow  cylinder,  called  a  nut,  having  a 
similar  spiral  within  it,  is  fitted  to  move  freely  upon  the  thread  of 
the  solid  cylinder.    In  Fig.  93,  let  the  base  A  B  of  the  inclined 
plane  A  0  be  equal  to  twice  the  circumference  of  the  cylinder 
A'  E\  then  let  the  plane  be  wrapped  about  the  cylinder,  bringing 
the  points  A,  F,  and  B,  to  the  point  A' ;  then  will  A  C  describe 
two  revolutions  of  the  thread  from  A'  to  C'.     Thei 


84 


MECHANICS. 


chanical  relations  of  the  screw  are  the  same  as  of  the  inclined 

plane. 

FIG.  93. 


If  a  weight  be  laid  on  the  thread  of  the  screw,  nud  a  force  be 
applied  to  it  horizontally  in  the  direction  of  a  tangent  to  the 
cylinder,  the  case  is  exactly  analogous  to  that  of  a  body  moved  on 
an  inclined  plane  by  a  force  parallel  to  the  base.  Let  r  be  the 
radius  of  the  cylinder,  then  2  TT  r  is  the  circumference ;  also  let  d 
be  the  distance  between  the  threads,  (that  is,  from  any  point  of 
one  revolution  to  the  corresponding  point  of  the  next,)  measured 
parallel  to  the  axis  of  the  cylinder ;  then  2  TT  r  is  the  base  of  an  in- 
clined plane,  and  d  its  height.  Therefore  (Art.  125), 
P  :  W::  d  :2n  r;  or, 

TJie power  is  to  the  weight  as  the  distance  between  the  threads 
measured  parallel  to  the  axis,  is  to  the  circumference  of  the  screw. 

If  instead  of  moving  the  weight  on  the  thread  of  the  screw,  the 
force  is  employed  to  turn  the  screw  itself,  while  the  weight  is  free 
to  move  in  a  vertical  direction,  the  law  is  the  same.  Thus, 
whether  the  screw  A'  E  is  allowed  to  rise  and  fall  in  the  fixed  nut 
G  H,  or  whether  the  nut  rises  and  falls  on  the  thread  of  the  screw, 
while  the  latter  is  revolved,  without  moving  longitudinally,  in 
each  case,  P  :  W  : :  d  :  2  rr  r. 

132.  The  Screw  and  Lever  Combined,— The  screw  is  so 
generally  combined  with  the  lever  in  practical  mechanics,  that  it 
is  important  to  present  the  law  of  the 
compound  machine.  Let  A  F  (Fig.  94) 
be  the  section  of  a  screw,  and  suppose 
B  C,  a  lever  of  the  second  order,  to  be 
applied  to  turn  it.  The  fulcrum  is  at  C, 
the  power  acts  at  B,  and  the  effect  pro- 
duced by  the  lever  is  at  A,  the  surface  of 
the  cylinder.  Call  that  effect  x,  and  let 
d  =  the  distance  between  the  threads : 


FIG.  94. 


THE    ENDLESS     SCREW.  85 

P  :x  \\AG\B  C, 
and  x  :  W  : :  d  :  2  TT  A  C', 
compounding  and  reducing,  we  have 

P  :  W::d:27T  B  C',  that  is, 

TJie  power  is  to  the  weight  as  the  distance  between  the  threads, 
measured  parallel  to  the  axis,  to  the  circumference  described  by  the 
power. 

The  law  as  thus  stated  is  applicable  to  the  screw  when  used 
with  the  lever  or  without  it. 

133.  The  Endless  Screw. — The  screw  is  so  called,  when 
its  thread  moves  between  the  teeth  of  a  wheel,  thus  causing  it  to 
revolve.    It  is  much  used  for  diminish- 
ing  very   greatly  the  velocity   of  the  'IG' 

weight. 

Let  P  Q  (Fig.  95)  be  the  radius  of 
the  crank  to  which  the  power  is  ap- 
plied; d,  the  distance  between  the 
threads ;  R,  the  radius  of  the  wheel ; 
r,  the  radius  of  the  axle;  and  call  the 
force  exerted  by  the  thread  upon  the 
teeth,  x.  Then,  wj 

P:x::d:2n  x  P  Q, 
and  x  :  W  : :  r  :  R ; 

.-.  P:  W  ::dr  :2  n  x  R  X  P  Q. 

If,  for  example,  P  Q  =  30  inches,  d  =  1  in.,  R  —  18  in. ; 
r  +  t  =  2  in.;  then  W  =  1696  P,  and  moves  with  1696  times 
less  velocity  than  P. 

134.  The  Right  and  Left  Hand  Screw. — The  common 
form  of  screw  is  called  the  right-hand  screw,  and  may  be  described 
thus  :  if  the  thread  in  its  progress  along  the  length  of  the  cylinder, 
passes  from  the  left  over  to  the  right,  it  is  called  a  right-hand  screw. 
Hence,  a  person  in  driving  a  screw  forward  turns  it  from  his  left 
over  (not  under)  to  his  right,  and  in  drawing  it  back  he  reverses 
this  movement.     Fig.  93  represents  a  right-hand  screw. 

The  left-hand  screw  is  one  whose  thread  is  coiled  in  the  oppo- 
site direction, — that  is,  it  advances  by  passing  from  right  over  to 
left.  This  kind  is  used  only  when  there  is  special  reason  for  it. 
For  example,  the  screws  which  are  cut  upon  the  left-hand  ends  of 
carriage  axles  are  left-hand  screws ;  otherwise  there  would  be 
danger  that  the  friction  of  the  hub  against  the  nut  might  turn 
the  nut  off  from  the  axle.  Also,  when  two  pipes  for  conveying 
gas  or  steam  are  to  be  drawn  together  by  a  nut,  one  must  have  a 
right-hand,  and  the  other  a  left-hand  screw. 


86  MECHANICS. 

135.  Questions  on  the  Screw. — 

1.  The  distance  between  the  threads  of  a  screw  is  one  inch, 
the  bar  is  two  feet  long  from  the  axis,  and  the  power  is  30  Ibs.; 
what  is  the  weight  or  pressure  ?  Ans.   4523.89  Ibs. 

2.  The  bar  is  three  feet  long,  reckoned  from  the  axis,  P  =  60 
Ibs.,  W  =  2240  Ibs.  ;   what  is  the  distance  between  the  threads? 

Ans.  6.058  inches. 

3.  A  compound  machine  consists  of  a  crank,  an  endless  screw, 
a  wheel  and  axle,  a  pulley,  and  an  inclined  plane.     The  radius  of 
the  crank  is  18  inches ;  the  distance  between  the  threads  of  the 
screw,  one  inch  ;  the  radius  of  the  wheel  on  which  the  screw  acts, 
two  feet ;  the  radius  of  the  axle,  6  inches ;  the  pulley  block  has 
two  movable  pulleys  with  one  rope,  the  power  exerted  by  the 
pulley  being  parallel  to  the  plane,  and  the  inclination  of  the  plane 
to  the  horizon  is  30°.     What  weight  on  the  plane  will  be  balanced 
by  a  power  of  100  Ibs.  applied  to  the  crank  ? 

Ans.  361911.474  Ibs. 

VII.  THE  WEDGE. 

136.  Definition   of  the  Wedge,   and    the    Mode    of 
Using. — The  usual  form  of  the  wedge  is  a.  triangular  prism,  two 
of  whose  sides  meet  at  a  very  acute  angle.     This  machine  is  used 
to  raise  a  weight  by  being  driven  as  an  inclined  plane  underneath 
it,  or  to  separate  the  parts  of  a  body  by  being  driven  between 
them.     When  it  is  used  by  itself,  and  does  not  form  part  of  a 
compound  machine,  force  is  usually  applied  by  a  blow,  which  pro- 
duces an  intense  pressure  for  a  short  time,  sufficient  to  overcome 
a  great  resistance. 

137.  Law  of  Equilibrium.— Whatever  be          FIG.  96. 
the  direction  of  the  blow  or  force,  we  may  sup- 
pose it  to  be  resolved  into  two  components,  one 
perpendicular  to  the  back  of  the  wedge,  and  the 

other  parallel  to  it.  The  latter  produces  no  effect. 
The  same  is  true  of  the  resistances  ;  we  need  to 
consider  only  those  components  of  them  which 
are  perpendicular  to  the  sides  of  the  wedge. 

Let  M NO  (Fig.  96)  represent  a  section  of 
the  wedge  perpendicular  to  its  faces  ;  then  P  A, 
Q  A,  and  R  A,  drawn  perpendicular  to  the  faces 
severally,  show  the  directions  of  the  forces  which 
hold  the  wedge  in  equilibrium.  Taking  A  B  to 
represent  the  power,  draw  B  C  parallel  to  R  A, 
and  we  have  the  triangle  A  B  C,  whose  sides 
represent  these  forces.  But  A  B  C  is  similar  to  M N 0, .as  their 


THE    KNEE-JOINT.  87 

sides  are  respectively  perpendicular  to  each  other.     Hence,  calling 
the  forces  P,  Q,  and  7?,  respectively, 

P  :Q::  M N:MOi 
and  P  :R'.\  M N :  NO; 
that  is,  there  is  equilibrium  in  a  wedge,  when 

The  power  is  to  the  resistances  as  the  back  of  the  wedge  to  the 
sides  on  which  the  resistances  respectively  act. 

If  the  triangle  is  isosceles,  the  two  resistances  are  equal,  as  the 
proportions  show  ;  and  P  is  to  either  resistance,  R,  as  the  breadth 
of  the  back  to  the  length  of  the  side. 

If  the  resisting  surfaces  touch  the  sides  of  the  wedge  only  in 
one  point  each,  then  Q  A  and  R  A,  drawn  through  the  points  of 
contact,  must  meet  A  P  in  the  same  point  (Art.  43)  ;  otherwise 
the  wedge  will  roll,  till  one  face  rests  against  the  resisting  body 
in  two  or  more  points. 

The  efficiency  of  the  wedge  is  usually  very  much  increased  by 
combining  its  own  action  with  that  of  the  lever,  since  the  point 
where  it  acts  generally  lies  at  a  distance  from  the  point  where  the 
effect  is  to  be  produced.  Thus,  in  splitting  a  log  of  wood,  the 
resistance  to  be  overcome  is  the  cohesion  of  the  fibers  ;  and  this 
force  is  exerted  at  a  distance  from  the  wedge,  while  the  fulcrum 
is  a  little  further  forward  in  the  solid  wood. 

VIII.  THE  KNEE-JOINT. 

138.  Description  and  Law  of  Equilibrium. — The  knee- 
joint  consists  of  two  bars,  usually  equal,  hinged  together  at  one 
end,  while  the  others  are  at  liberty  to  separate  in  a  straight  line. 
The  power  is  applied  at  the  hinge,  tending  to  thrust  the  bars 
into  a  straight  line ;  the  weight  is  the  force  which  opposes  the 
separation. 


TV- 


Suppose  that  A  B  and  A  D  (Fig.  97)  are  equal  bars,  hinged 
together  at  A  ;  and  that  the  bar  A  B  is  free  only  to  revolve  about 


88 


MECHANICS. 


the  axis  B,  while  the  end  D  of  the  other  bar  can  move  parallel  to 
the  base  E  F.  If  P  urges  A  toward  the  base,  it  tends  to  move  D 
further  from  the  fixed  point  B.  The  force  P',  which  opposes 
that  motion,  is  represented  in  the  figure  by  the  weight  W.  The 
law  of  equilibrium  is, 

The  power  is  to  the  weight  as  twice  the  height  of  the  joint  to 
half  the  distance  between  the  ends  of  the  bars. 


m 

Resolving  the  force  P  in  the  direction  of  A  B  and  A  /),  we 
have,  Fig.  98, 

P  :T::AH:AG  ::2AC:AD, 

in  which  T  stands  for  the  component  of  P  in  the  direction  A  G, 
called  the  thrust. 

This  component  T  acts  at  D  and  must  be  again  resolved  in 
the  directions  D  L  and  D  M,  of  which  D  L  is  equal  and  opposed 
to  W,  and  D  M  is  equal  and  opposed  to  the  upward  resistance  of 
the  plane  on  which  the  block  D  slides,  giving  the  proportion 

T:  W'.'.DK-.D  L  or  M K-.\  A  D  \  C D. 

Multiplying  like  terms  of  the  two  proportions  and  omitting  com- 
mon factors,  we  have, 

P\  W::  2  A  0:  CD. 

139.   Ratio   of   Power   and  Weight  Variable.— It  is 

obvious  that  the  ratio  between  power  and  weight  is  different  for 
different  positions  of  the  bars.  As  A  is  raised  higher  C  D  dimin- 
ishes, and  when  the  bars  are  parallel,  we  have 

P:  W::2  A  (7:0; 


VIRTUAL    VELOCITIES.  8U 

that  is  to  say,  the  power  has  no  efficiency.  But  as  A  approaches 
the  base  A  C  diminishes,  and  at  last  we  have,  when  B  A  and 
A  D  are  in  the  same  line, 

P:  W  ::0:  B  A. 

Hence  the  weight  or  resistance  in  such  case  is  infinite  as  com- 
pared with  the  power  applied.  The  indefinite  increase  of  effi- 
ciency in  the  power,  which  occurs  during  a  single  movement, 
renders  this  machine  one  of  the  most  useful  for  many  purposes, 
as  printing  and  coining. 

Questions  on  the  knee-joint. — 

1.  A  power  of  50  Ibs.  is  exerted  on  the  joint  A  (Fig.  97) ; 
compare  the  weight  which  will  balance  it,  when  B  A  D  is  90°, 
and  when  it  is  160°.  Ans.  25  Ibs.  and  141.78  Ibs. 

2.  When  the  angle  between  the  bars  is  110°,  a  certain  power 
just  overcomes  a  weight  of  65  Ibs. ;  what  must  be  the  angle,  in 
order  that  the  weight  overcome  may  be  five  times  as  great  ? 

Ans.  164°  3'  22". 

PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

140.  Definition. — The  virtual  velocity  of  a  point,  with  respect 
to  any  force,  is  the  product  of  its  actual  velocity  by  the  cosine  of 
the  angle  which  its  actual  path  makes  with  the  direction  of  the 
force.     Thus,  let  a  point  A  (Fig.  99)  be  acted  upon  by  a  force 
P  in  the  direction  A  c,  and  be- 
cause of  some  other  external  force 

or  resistance  suppose  the  point 
to  be  constrained  to  move  in  the 
line  A  A'  to  A'  in  any  unit  of 
time  :  then  A  d,  the  projection 
of  A  A'  upon  A  c,  is  the  virtual 
velocity  of  the  point  A  with  reference  to  the  force  P. 

141.  The  Point  of  Application  Moving  in  the  Line  of 
the  Force. — It  can  be  shown,  in  every  case,  that  the  velocities, 
when  reckoned  in  the  direction  in  which  the  forces  act,  are  inversely 
as  the  forces. 

Some  examples  are  first  given  in  which  the  point  of  applica- 
tion moves  in  the  line  in  which  the  force  acts. 

In  the  straight  lever  (Fig.  100),  which  is  in  equilibrium  by  the 

weights  P  and  W,  suppose  a 

FIG.  100.  slight  motion  to  exist ;  then 

the  velocity  of  each  will  be 
as  the  arc  described  in  the 
same  time  ;  but 'the  arcs  are 
similar,  since  they  subtend 


90  MECHANICS. 

equal  angles.  Therefore,  if  F  =  velocity  of  P,  and  v  =  velocity 
of  W. 

V  :v::A  P  :  B  W::A  C :  B  (7; 
but  it  has  been  shown  (Art.  106)  that 

P:  W::B  O:A  (7; 
.*.    F  :  v  ::    W   :   P; 

that  is,  the  velocity  of  the  power  is  to  the  velocity  of  the  weight 
as  the  weight  to  the  power.  Hence,  P  x  its  velocity  =  W  x  its 
velocity ;  that  is,  the  momentum  of  the  power  equals  the  mo- 
mentum of  the  weight. 

In  the  wheel  and  axle,  let  R  and  r  he  the  radii,  and  suppose 
the  machine  to  be  revolved ;  then  while  P  descends  a  distance 
equal  to  the  circumference  of  the  wheel  =  2  TT  R9  the  weight 
ascends  a  distance  equal  to  the  circumference  of  the  axle  =  2  n  r. 
Therefore, 

V  :V::%TT  R  :  2  n  r::  R  :  r; 
but  (Art.  113),  P  :  W : :  r  :  R ; 

.-.  Vivii  W  :  P; 

or,  the  velocities  are  inversely  as  the  weights  ;  and  P  x  F=  JFx  v, 
the  momentum  of  the  power  equals  the  momentum  of  the 
weight. 

In  the  fixed  pulley  the  velocities  are  obviously  equal ;  and  we 
have  before  seen  that  the  power  and  weight  are  equal;  therefore 
the  proportion  holds  true,  F  :  v  : :  W  :  P ;  and  the  momenta  are 
equal. 

In  the  movable  pulley,  if  n  is  the  number  of  sustaining  parts 
of  the  cord,  when  W  rises  any  distance  =  x,  each  portion  of  cord 
is  shortened  by  the  distance  x,  and  all  these  n  portions  pass  over 
to  P,  which  therefore  descends  a  distance  =  n  x. 

Hence, 

V  :  v  : :  n  x  :  x  : :  n  :  1 ; 
but  (Art.  120),  P  :  W ::  1  :  w; 

/.  V:v::  W:P\ 
as  in  all  the  preceding  cases. 

In  the  screw  (Fig.  94),  while  the  power  describes  the  circum- 
ference =  2  n  x  B  C,  the  weight  moves  only  the  distance  =  d ; 
therefore, 

F:  v::27r  x  B  C:d; 
but  (Art.  131),          P:  W::d:%TT  x  B  (7; 
.-.  V-.vi:  W:  P\ 

therefore  the  momentum  of  the  power  equals  the  momentum  of 
the  weight,  as  before. 


VIRTUAL    VELOCITIES. 


01 


FIG.  101. 


142.  The  Point  of  Application  Moving  in  a  Different 
Line  from  that  in  which  the  Force  Acts. — The  cases  thus 
far  noticed  are  the  most  obvious  ones,- because  the  points  of  appli- 
cation of  power  and  weight  actually  move  in  the  directions  in 
which  their  force  is  exerted.  The  case  of  the  inclined  plane  will 
illustrate  the  principle,  when  the  point  of  application  does  not 
move  in  the  direction  of  the  force. 

First,  let  P  (Fig.  101)  act  parallel  to  the  plane,  and  suppose 
the  body  to  be  moved  either  up  or  down  the  plane  a  distance  equal 
to  G  d.  That  is  the  velocity 
of  the  power.  But  in  the  di- 
rection of  the  iveight  (force  of 
gravity)  the  body  moves  only 
the  distance  b  cl  Therefore 
the  velocity  of  the  power  is  to 
the  velocity  of  the  weight 
(each  being  reckoned  in  the 
line  of  its  action)  as  G  d  to 
Id. 

By  similar  triangles,     G  d  :b  d  : :  A  C  :  A  B; 
or     V :   v   : :  A  C :  A  B. 

But  (Art.  125),  Pi   W::ABiAC', 

.-.     V:   v   : :     W  :  P. 

Again,  let  the  power  act  in  any  oblique  direction,  as  G  e.  If 
the  body  moves  over  G  d,  draw  d  e  perpendicular  to  G  e ;  then  G  e 
is  the  distance  passed  over  in  the  direction  of  the  power,  and  b  d  in 
the  direction  of  the  weight.  G  d  being  taken  as  radius,  G  e  is 
cos  d  G  e  =  cos  (P  G  N  —  90°)  —  sin  P  G  N;  and  b  d=sin  a. 
Therefore,  the  virtual  velocity  of  the  power  is  to  the  virtual  velo- 
city of  the  weight  as  sin  P  G  N  to  sin  a  ; 

or  V :  v  : :  sin  P  G  N :  sin  a. 
But  (Art.  126),  P  :  W  : :  sin  a  :  sin  P  G  N] 

.:  Viv.i  W:  P. 

We  learn  from  the  foregoing  principle,  that  a  machine  does  not 
enable  us  to  obtain  any  greater  effect  than  the  power  could  pro- 
duce without  its  aid,  but  only  to  produce  an  effect  in  a  different 
form.  A  given  power,  for  instance,  may  move  a  much  greater 
quantity  of  matter  by  the  aid  of  a  machine,  but  it  will  move  it  as 
much  more  slowly.  On  the  other  hand,  a  power,  by  means  of  a 
machine,  may  produce  a  far  greater  velocity  than  would  be  pos- 
sible without  such  aid  ;  but  the  quantity  moved,  or  the  intensity 
of  the  force  exerted,  would  be  proportionally  less.  By  machines, 
therefore,  we  do  not  increase  the  effects  of  a  power,  but  only 
modify  them. 


93  MECHANICS. 

FBICTION  IK  MACHINERY. 

143.  The  Power  and  Weight  not  the  only  Forces  in 
a  Machine. — For  each  machine  a  certain  proportion  has  been 
given,  which  ensures  equilibrium.     And  it  is  implied  that  if  either 
the  power  or  the  weight  be  altered,  the  equilibrium  will  be  de- 
stroyed.    But  practically  this  is  not  true ;  the  power  or  weight 
may  be  considerably  changed,  or  possibly  one  of  them  may  be 
entirely  removed,    and  the  machine  still  remain  at  rest.     The 
obstruction    which  prevents  motion  in  such   cases,  and  which 
always  exists  in  a  greater  or  less  degree,  arises  from  friction  ;  and 
friction  is  caused  by  roughness  in  the  surfaces  which  rub  against 
each  other.    The  minute  elevations  of  one  surface  fall  in  between 
those  of  the  other,  and  directly  interfere  with  the  motion  of 
either,  while  they  remain  in  contact.     Polishing  diminishes  the 
friction,  but  can  never  remove  it,  for  it  never  removes  all  rough- 
ness. 

The  coefficient  of  friction  is  the  fraction  whose  numerator  is  the 
force  required  to  overcome  the  friction,  and  its  denominator  the 
normal  pressure  betiveen  the  bodies. 

In  the  case  of  bodies  whose  surfaces  of  contact  are  horizontal 
the  denominator  in  the  coefficient  is  the  weight  of  the  pressing 
body. 

As  friction  always  tends  to  prevent  motion,  and  never  to  pro- 
duce it,  it  is  called  a  passive  force.  It  assists  the  power,  when 
the  weight  is  to  be  kept  at  rest,  but  opposes  it,  when  the  weight 
is  to  be  moved.  There  are  other  passive  forces  to  be  considered 
in  the  study  of  science,  but  no  other  has  so  much  influence  in  the 
operations  of  machinery  as  friction. 

144.  Modes  of  Experimenting. — When  one  surface  slides 
on  another,  the  friction  which  exists  is  called  the  sliding  friction  ; 
but  when  a  wheel  rolls  along  a  surface,  the  friction  is  called  roll- 
ing friction.     The  sliding  friction  occurs  much  more  in  machines 
than  the  rolling  friction. 

Experiments  for  ascertaining  the  laws  of  friction  may  be  per- 
formed by  placing  on  a  table  a 
block  of  three  different  dimen- 
sions, and  measuring  its  friction 
under    different   circumstances 
by  weights  acting  on  the  block 
by  means  of  a  cord  and  pulley, 
as  represented  in  Fig.  102.   This 
was  the  method  by  which  Coulomb  first  ascertained  the  laws 
of  friction. 


FRICTION    IN    MACHINERY.  93 

Another  mode  is  to  place  the  block  on  an  inclined  plane,  whose 
angle  can  be  varied,  and  then  find  the  relative  friction  in  different 
cases,  by  the  largest  inclination  at  which  it  will  prevent  the  block 
from  sliding.  For,  when  W  on  the  inclined  plane  A  B  (Fig.  103) 
is  on  the  point  of  sliding  down,  F 

friction  is  the  power  which,  acting 
parallel  to  the  plane,  is  in  equili- 
brium with  that  component  of  the 
weight  which  tends  to  move  the 
block  down  the  plane. 

This  component  parallel  to  the 
plane,  is  IF  sin  A,  and  the  normal 
pressure  W  cos  A  ;  hence,  calling  the  coefficient  of  friction  u,  we 

have  (Art.  143)  u  =  -^ -r  =  tan  A,  or  the  coefficient  of  fric- 
tion is  equal  to  the  tangent  of  the  angle  of  inclination  of  the  plane. 
For  example,  suppose  a  block  of  cast  iron  to  rest  upon  an  oak 
plank,  and  that  the  end  of  the  plank  is  raised  so  that  the  block 
slides  with  uniform  motion  down  the  plane  ;  then  the  angle  A 
will  be  found  by  actual  measurement  to  be  about  26°,  the  natural 
tangent  of  which  is  .48773  ;  hence,  in  pounds,  49  per  cent,  of  the 
weight  will  represent  the  force  in  pounds  required  to  overcome 
the  friction. 

145.  Laws  of  Sliding  Friction. — The  laws  of  sliding 
friction  on  which  experimenters  are  generally  agreed  are  the 
following : 

1.  Friction  varies  as  the  pressure. — If  weights  are  put  upon 
the  block,  it  is  found  that  a  double  weight  requires  a  double  force 
to  move  it,  a  triple  weight  a  triple  force,  &c. 

2.  It  is  the  same,  however  great  or  small  the  surface  on  which 
the  body  rests. — If  the  block  be  drawn,  first  on  its  broadest  side, 
then  on  the  others  in  succession,  the  force  required  to  overcome 
friction  is  found  in  each  case  to  be  the  same.     Extremes  of  size 
are,  however,  to  be  excepted.     If  the  loaded  block  were  to  rest 
on  three  or  four  very  small  surfaces,  the  obstruction  might  be 
greatly  increased  by  the  indentations  thus  occasioned  in  the  sur- 
face beneath  them. 

3.  Friction  is  a  uniformly  retarding  force. — That  is,  it  destroys 
equal  amounts  of  motion  in  equal  times,  whatever  may  be  the 
velocity,  like  gravity  on  an  ascending  body. 

4.  Friction  at  the  first  moment  of  contact  is  less  than  after  con- 
fact  has  continued  for  a  time. — And  the  time  during  which  fric- 
tion increases,  varies  in  different  materials.    The  friction  of  wood 
on  wood  reaches  its  maximum  in  three  or  four  minutes  ;  of  metal 


94  MECHANICS. 

on  metal,  in  a  second  or  two  ;  of  metal  on  wood,  it  increases  for 
several  days.  As  any  jar  or  vibration  changes  at  once  the  friction 
of  rest  to  that  of  motion,  the  coefficients  to  be  considered  in 
determining  the  stability  of  any  vstructure  should  be  those  of 
motion. 

5.  Friction  is  less  between  substances  of  different  kinds  than 
bettveen  those  of  the  same  kind. — Hence,  in  watches,  steel  pivots 
are  made  to  revolve  in  sockets  of  brass  or  of  jewels,  rather  than 
of  steel. 

146.  Friction  of  Axes. — In  machinery,  the  most  common 
case  of  friction  is  that  of  an  axis  revolving  in  a  hollow  cylinder,  or 
the  reverse,  a  hollow  cylinder  revolving  on  an  axis.     These  are 
cases  of  sliding  friction,  in  which  the  power  that  overcomes  the 
friction,  usually  acts  at  the  circumference  of  a  wheel,  and  there- 
fore at  a  mechanical  advantage.     Thus,  the  friction  on  an  axis, 
whose  coefficient  is  as  high  as  20  per  cent.,  requires  a  power  of 
only  two  per  cent,  to  overcome  it,  provided  the  power  acts  at  the 
circumference  of  a  wheel  whose  diameter  is  ten  times  that  of 
the  axis. 

147.  Rolling  Friction. — This  form  of  friction  is  very  much 
less  than  the  sliding,  since  the  projecting  points  of  the  surfaces 
do  not  directly  encounter  each  other,  but  those  of  the  rolling 
wheel  are  lifted  up  from  among  those  of  the  other  surface,  as  the 
wheel  advances. 

By  the  use  of  the  apparatus  described  in  Art.  144,  the  laws  of 
the  rolling  are  found  to  be  the  same  as  those  of  the  sliding  friction. 
But  on  account  of  the  manner  in  which  this  form  of  friction  is 
overcome,  there  is  this  additional  law : 

TJie  force  required  to  roll  the  wheel  varies  inversely  as  the 
diameter. 

For  the  power,  acting  at  the  centre  of  the  wheel  to  turn  it  on 
its  lowest  point  as  a  momentary  fulcrum,  has  the  advantage  of 
greater  acting  distance  as  the  diameter  increases. 

It  is  the  rolling  friction  which  gives  value  to  friction  wheels, 
as  they  are  called.  When  it  is  desirable  that  a  wheel  should 
revolve  with  the  least  possible  friction,  each  end  of  its  axis  is  made 
to  rest  in  the  angle  between  two  other  wheels  placed  side  by  side, 
as  shown  in  Fig.  104.  The  wheel  is  obstructed  only  by  the  rolling 
friction  on  the  surfaces  of  the  four  wheels,  and  the  retarding  effect 
of  the  sliding  friction  at  the  pivots  of  the  latter  is  greatly  reduced 
on  the  principle  of  the  wheel  and  axle. 

The  sliding  friction  is  diminished  by  lubricating  the  surface, 
tne  rolling  friction  is  not. 


LIMITING    ANGLE   OF    FRICTION.  95 

FIG,  104. 


148.  Advantages  of  Friction. — Friction  in  machinery  is 
generally  regarded  as  an  evil,  since  more  power  is  on  this  account 
required  to  do  the  work  for  which  the  machine  is  made.  But  it 
is  easy  to  see,  that  in  general  friction  is  of  incalculable  value,  or 
rather,  that  nothing  could  be  accomplished  without  it.  Objects 
stand  firmly  in  their  places  by  friction  ;  and  the  heavier  they  are, 
the  more  firmly  they  stand,  because  friction  increases  with  the 
pressure.  All  fastening  by  nails,  bolts,  and  screws,  is  due  to  fric- 
tion. The  fibers  of  cotton,  wool  or  silk,  when  intertwined  with 
each  other,  form  strong  threads  or  cords,  only  because  of  the  power 
of  friction.  Without  friction,  it  would  be  impossible  to  walk  or 
even  to  stand,  or  to  hold  anything  by  grasping  it  with  the  hand. 


149.  Limiting  Angle  of  Friction.— If 

contact  and  forces  be  applied,  oblique  to  the 
no  motion  will  result  so  long  as  the  resultant 
with  the  normal  an  angle  whose  tangent  is  less 
of  friction,  no  matter  how  great  this  resultant 
Thus  (Fig.  105)  let  a  block  of  iron  rest 
oak,  as  in  the  case  heretofore  considered,  and 
let  the  force  P  be  applied.  In  this  case  the 
forces  are  P  and  W,  and  their  resultant  is  R 
(or  R')  which  may  be  considered  in  their 
stead.  The  component  of  R1 ',  which  tends  to 
produce  motion,  is  R'  sine  a,  and  the  total 

,„  R1  sine  a 

normal  pressure  is  R  cos  a.     If  -=r, —     -  = 

R  cos  a 

tan  a,  is  less  than  the  coefficient  of  friction  no 


two  surfaces  are  in 
surface  of  contact, 
of  all  forces  makes 
than  the  coefficient 
force  may  be. 
upon  a  surface  of 

FIG.  105. 


90  MECHANICS. 

motion  can  result ;  that  is  to  say,  if  a  is  less  than  the  inclination 
of  the  plane,  in  Art.  144,  there  will  be  stable  equilibrium. 

The  greatest  angle  a  which  the  resultant  of  all  the  forces  can 
make  with  the  normal  without  producing  sliding  motion  of  the 
surfaces  is  called  the  limiting  angle  of  friction,  and  its  tangent  is 
equal  to  the  coefficient  of  friction. 

[NOTE. — The  laws  of  sliding  friction  have  claimed  the  attention  of  modern 
experimenters,  and  the  results  obtained  modify  very  essentially  the  laws  given 
in  the  text.  Prof.  A.  S.  Kimball,  of  Worcester,  gives  the  following  conclu- 
sions drawn  from  his  numerous  experiments :  "  The  coefficient  of  friction  at 
very  low  velocities  is  small ;  it  increases  rapidly  at  first,  then  more  gradually 
as  the  velocity  increases,  until  at  a  certain  rate,  which  depends  upon  the 
nature  of  the  surfaces  in  contact  and  the  intensity  of  the  pressure,  a  maxi- 
mum coefficient  is  reached.  As  the  velocity  increases  beyond  this  point  the 
coefficient  decreases." 

"  For  a  considerable  range  of  velocities  in  the  vicinity  of  the  maximum 
coefficient  the  coefficient  is  sensibly  constant." 

Prof.  R.  H.  Thurston,  of  Hoboken,  draws  the  following  conclusions  from 
his  experiments  upon  friction  of  lubricated  journals  :  "  Studying  table  A  we 
see  that  the  coefficient  rapidly  diminishes  with  increase  of  pressure,  until  a 
pressure  of  500  Ibs.  per  square  inch  is  attained  ;  the  coefficient,  after  passing 
a  pressure  of  probably  600  to  800  Ibs.  per  square  inch,  increases,  and,  at  1000 
Ibs.,  becomes  about  equal  to  that  obtained  at  100  Ibs." 

"The  coefficients  of  quiescence  increase  with  the  pressure,  instead  of 
diminishing  as  do  the  coefficients  of  friction  of  motion  ;  and  at  the  highest 
pressures,  their  values  become  from  ten  to  forty  times  the  corresponding 
values  of  the  latter." 

The  results  of  all  the  more  modern  experiments  must  be  collated  before 
the  laws  of  friction  can  be  given  satisfactorily.] 


CHAPTER    VII. 

MOTION  ON  INCLINED  PLANES.— THE  PENDULUM. 

150.  The  Force  which  Moves  a  Body  Down  an  In- 
clined Plane. — It  was  shown  (Art.  125)  that  when  the  power 
acts  in  a  line  parallel  to  the  inclined  plane,  P  :  W : :  A  B  :  A  C. 
If,  therefore,  P  ceases  to  act,  the  body  descends  the  plane  only 
with  a  force  equal  to  P. 

Let  g  (the  velocity  acquired  in  a  second  in  falling  freely)— the 
force  of  gravity,  /=  the  force  acting  down  the  plane,  A=the 
height,  I  =  the  length ;  then  by  substitution, 
/  :  g  : :  h  :  19  and 

/-* 


FORMULAE    FOR    THE    INCLINED    PLANE.         97 

Therefore,  the  force  which  moves  a  body  down  an  inclined 
plane  is  equal  to  that  fraction  of  gravity  which  is  expressed  by 
the  height  divided  by  the  length.  This  is  evidently  a  constant 
force  on  any  given  plane,  and  produces  uniformly  accelerated 
motion.  Therefore  the  motion  on  an  inclined  plane  does  not 
differ  from  that  of  free  fall  in  kind,  but  only  in  degree.  Hence 
the  formulas  for  time,  space,  and  velocity  on  an  inclined  plane  are 
like  those  relating  to  free  fall,  if  the  value  of  /  be  substituted 

151.  Formulae  for  the  Inclined  Plane. — The  formulae 
for  free  fall  (Art.  27)  are  here  repeated,  and  against  them  the 
corresponding  formulae  for  descent  on  an  inclined  plane. 

Free  fall.  Descent  on  an  inclined  plane. 

1.      ,  =  *—  —!>** 


2.      »>4/H 

9 

_v> 

S-      S-*~9    • 


V  = 


5.  *  = 

9 

6.  v  = 


By  formula  1,  s  oc  £2,  and  by  formula  3,  s  oo  v*.  It  follows  that 
in  equal  successive  times  the  spaces  of  descent  are  as  the  odd 
numbers,  1,  3,  5,  &c.,  and  of  ascent  as  these  numbers  inverted; 
also,  that  with  the  acquired  velocity  continued  uniformly,  a  body 
moves  twice  as  far  as  it  must  descend  to  acquire  that  velocity.  If 
a  body  be  projected  up  an  inclined  plane,  it  will  ascend  as  far  as  it 
must  descend  in  order  to  acquire  the  velocity  of  projection.  The 
distance  passed  over  in  the  time  t  by  a  body  projected  with  the 

velocity  v,  down   or  up  an  inclined  plane,  equals  t  v  ±  ^r~r- 

A  I 

152.  Formulae  for  the  whole  Length  of  a  Plane.  — 

1.  The,  velocity  acquired  in  descending  a  plane  is  tlie  same  as 
that  acquired  in  falling  down  its  height. 

For  now  s  =  I  ;  hence  (formula  4),  v  —  (—  —  —  )=:  (  2  g  h  )*, 

which   is  the  formula  for  free  fall  through  h,  the  height  of  the 
plane. 

7 


MECHANICS. 


On  different  planes,  therefore,  v  QC  h*. 

2.  The  time  of  descending  a  plane  is  to  the  time  of  fatting  down 
its  height  as  the  length  to  the  height. 

For  (formula  2)  *=:(^)  =  *(--).      But  the  time  of  fall 


down  the  height  is  (— j.     Therefore, 

t  down  plane  :  t  down  height  : :  I  \—j\  '•  ( — )  ; 


On  different  planes,  t  oc  -— 


:  :  I  :  h. 


It  follows  that  if  several  planes  have  the  same  height,  the  ve- 
locities acquired  in  descending  them  are  equal,  and  the  times  of 
descent  are  as  the  lengths  of  the  planes.  For,  let  A  C,  A  D,  A  E, 

(Fig.  106)  have  the  same  height  A  B',  then,  since  v  cc  k*9  and  h 
is  the  same  for  all,  v  is  the  same.  And  since  t  oc  —  —  ,  and  h  is 
the  same  for  all  the  planes,  t  x  I. 


FIG.  106. 


E  D        0  B  B 

153.  Descent  on  the  Chords  of  a  Circle.— In  descending 
the  chords  of  a  circle  which  terminate  at  the  ends  of  the  vertical 
diameter,  the  acquired  velocities  are  as  the  lengths,  and  the  times 
of  descent  are  equal  to  each  other  and  to  the  time  of  falling  through 
the  diameter. 

For  (Art.   152)   the  velocity  acquired  on  A  C  (Fig.  107)  = 

(2  ,  -  A.}=  (*,  4S)  *=  Ao(*g*.  which,  since 
is  constant,  varies  as  A  C,  the  length. 


NO    LOSS    ON    A    CURVE.  99 

Again   (Art.    152),   the    time    down    A    O=( -r- )   = 

\g  •  A.  c' 

f—      ~ j    ~\          )  '  which  is  equal  to  the  time  of  falling 

freely  through  A  B,  the  diameter. 

154.  Velocity  Acquired  on  a  Series  of  Planes. — If  no 

velocity  be  lost  in  passing  from  one  plane  to  another,  the  velocity 
acquired  in  descending  a  series  of  planes  is  equal  to  that  acquired 
in  falling  through  their  perpendicu-  FIG.  108. 

lar  height.     For,  in   Fig.  108,  the  A    E  F 

velocity  at  B  is  the  same,  whether 
the  body  comes  down  A  B  or  E  B, 
as  they  are  of  the  same  height,  F  b. 
If,  therefore,  the  body  enters  on  £  C 
with  the  acquired  velocity,  then  it  is 
immaterial  whether  the  descent  is 
on  A  B  and  B  C  or  on  E  (7;  in  »  G 

either  case,  the  velocity  at  C  is  equal  to  that  acquired  in  falling  Fc. 
In  like  manner,  if  the  body  can  change  from  B  (7  to  G  D  without 
loss  of  velocity,  then  the  velocity  at  D  is  the  same,  whether  ac- 
quired on  A  B,  B  (7,  and  C  D,  or  on  F  D,  which  is  the  same  as 
down  F  G. 

155.  The  Loss  in  Passing  from  one  Plane  to  Another. 

— The  condition  named  in  the  foregoing  article  is  not  fulfilled.  A 
body  does  lose  velocity  in  passing  from  one  plane  to  another.  And 
the  loss  is  to  the  whole  previous  velocity  as  the  versed  sine  of  the 
angle  between  the  planes  to  radius. 

Let  B  F  (Fig.  109)  represent  the  velocity  which  the  body  has 
at  B.  Resolve  it  into  B  D  on  the  second  plane,  and  D  ^perpen- 
dicular to  it.  B  D  is  the  initial  velocity  on  B  C; 
and,  if  B  I  =  B  F,  D  /is  the  loss.  But  D  I  is 
the  versed  sine  of  the  angle  F  B  D,  to  the  radius 
B  F\  and  /.  the  loss  is  to  the  velocity  at  B  as  D  I 
:  B  F : :  ver.  sin  B  :  rad. 

156.  No  Loss  on  a  Curve.— Suppose  now 
the  number  of  planes  in  a  system  to  be  infinite  ; 
then  it  becomes  a  curve  (Fig.  110).     As  the  angle 
between  two  successive  elements  of  the  curve  is  in- 
finitely small,  its  chord  is  also  infinitely  small ;  but 
its  versed  sine  is  infinitely  smaller  still,  i.  e.,  an  infi- 
nitesimal of  the  second  order  ;   for  diam.  :  chord  : :  chord  :  ver. 
sin.     Therefore,  although  the  sum  of  all  the  infinitely  small  angles 


100 


MECHANICS. 


is  a  finite  angle  180°  —  A  G  D,  yet,  as  the  loss  of  velocity  at  each 
point  is  an  infinitesimal  of  the  second  order,         FIG.  110.  A 

the  entire  loss  (which  is  the  sum  of  the  losses 
at  all  points  of  the  curve)  is  an  infinitesimal 
of  the  first  order. 

Hence,  a  body  loses  no  velocity  on  a  curve, 
and  therefore  acquires  at  the  bottom  the 
same  velocity  as  in  falling  freely  through  its 
height.  D 

It  appears,  therefore,  that  whether  a  body  descends  vertically, 
or  on  an  inclined  plane,  or  on  a  curve  of  any  kind,  the  acquired 
velocity  is  the  same,  if  the  height  is  the  same. 

157.  Times  of  Descending  Similar  Systems  of  Planes 
and  Similar  Curves. — If  planes  are  equally  inclined  to  the  hori- 
zon, the  times  of  describing  them  are  as  the  square  roots  of  their 
lengths.  For,  if  the  height  and  base  of  each  plane  be  drawn,  simi- 
lar triangles  are  formed,  and  h  :  I  is  a  constant  ratio  for  the  several 


planes.     By  Art.  152,  t  oc  — — -  oc  — —  oc 


that  is,  the  time 


varies  as  the  square  root  of  the  length. 

If  two  systems  of  planes  are  similar,  i.  e.,  if  the  correspond- 
ing parts  are  proportional  and.  equally  inclined  to  the  horizon,  it 
is  still  true  that  the  times  of  descending  them  are  as  the  square 
roots  of  their  lengths. 

LeiABCD  and  abed  (Fig.  Ill)  be  similar,  and  let  A  F  and 


a  e    f 


a  /be  drawn  horizontally,  and  the  lower  planes  produced  to  meet 
them,  then  it  is  readily  proved  that  all  the  homologous  lines  of 
the  figures  are  proportional,  and  their  square  roots  also  propor- 
tional. Then  (reading  t,  A  B,  time  down  A  B,  &c.), 


we  have 


t,  A  B  :  t,  a  b  :  : 
t,  EB  :  t,  eb  :: 


B  :      a  b  ; 

\  Ve~b::  \/~A~B 


THE    PENDULUM.  1D1 

and  t,  E  C:  t,  e  c  : :  VE~C :  Ve~c  : :  VAB  :  Va~b'f 

.-.  (by  subtraction)   /,  B  C :  t,  b  c  : :  V 'A  B  :  Va  b. 
In  like  manner,  t,  CD  :t,cd:\  V 'A  B  :  Va  b. 
.'.  (by  addition) 

t,  (AB  +  BC  +  CD]  :t,  (ab  +  bc  +  cd)  ::^TB:^'al 

: :  ^/(A  B  +  B  G  +  CD)  :  ^/(a  b  +  b  c  +  c  d). 

Though  there  is  a  loss  of  velocity  in  passing  from  one  plane 
to  another,  the  proposition  is  still  true  ;  because,  the  angles  being 
equal,  the  losses  are  proportional  to  the  acquired  velocities  ;  and 
therefore  the  initial  velocities  on  the  next  planes  are  still  in  the 
same  ratio  as  before  the  losses ;  hence  the  ratio  of  times  is  not 
changed. 

The  reasoning  is  applicable  when  the  number  of  planes  in  each 
system  is  infinitely  increased,  so  that  they  become  curves,  similar, 
and  similarly  inclined  to  the  horizon.  Suppose  these  curves  to  be 
circular  arcs ;  then,  as  they  are  similar,  they  are  proportional  to 
their  radii.  Hence,  the  times  of  descending  similar  circular  arcs 
are  as  the  square  roots  of  the  radii  of  those  arcs. 

158.  Questions  on  the  Motions  of  Bodies  on  Inclined 
Planes.— 

1.  How  long  will  it  take  a  body  to  descend  100  feet  on  a  plane 
whose  length  is  150  feet,  and  whose  height  is  60  feet  ? 

Ans.  3.9  sec. 

2.  There  is  an  inclined  railroad  track,  2£  miles  long,  whose 
inclination  is  1  in  35.     What  velocity  will  a  car  acquire,  in  run- 
ning the  whole  length  of  the  road  by  its  own  weight  ? 

Ans.  106.2  miles  per  hour. 

3.  A  body  weighing  5  Ibs.  descends  vertically,  and  draws  a 
weight  of  6  Ibs.  up  a  plane  whose  inclination  is  45°.     How  far  will 
the  first  body  descend  in  10  seconds  ?  Ans.  110.74  feet. 

159.  The  Pendulum. — A  pendulum  is  a  weight  attached 
by  an  inflexible  rod  to  a  horizontal  axis  of  suspension,  so  as  to  be 
free  to  vibrate  by  the  force  of  gravity.     If  it  is  drawn  aside  from 
its  position  of  rest,  it  descends,  and  by  the  momentum  acquired, 
rises  on  the  opposite  side  to  the  same  height,  when  gravity  again 
causes  its  descent  as  before.     If  unobstructed,  its  vibrations  would 
never  cease. 

A  single  vibration  is  the  motion  from  the  highest  point  on  one 
side  to  the  highest  point  on  the  other  side.  The  motion  from 
the  highest  point  on  one  side  to  the  same  point  again  is  called  a 
double  vibration. 


102 


MECHANICS. 


The  axis  of  the  pendulum  is  a  line  drawn  through  its  centre  of 
gravity  perpendicular  to  the  horizontal  axis  about  which  the  pen- 
dulum vibrates. 

The  centre  of  oscillation  of  a  pendulum  is  that  point  of  its  axis 
at  which,  if  the  entire  mass  were  collected,  its  time  of  vibration 
would  be  unchanged. 

The  length  of  a  pendulum  is  that  part  of  its  axis  which  is 
included  between  the  axis  of  suspension  and  the  centre  of  oscil- 
lation. 

All  the  particles  of  a  pendulum  may  be  conceived  to  be  col- 
lected in  points  lying  in  the  axis.  Those  which  are  above  the  cen- 
tre of  oscillation  tend  to  vibrate  quicker  (Art.  157),  and  therefore 
accelerate  it  ;  those  which  are  below  tend  to  vibrate  slower,  and 
therefore  retard  it.  But,  according  to  the  definition  of  the  centre 
of  oscillation,  these  accelerations  and  retardations  exactly  balance 
each  other  at  that  point. 

160.  Calculation  of  the  Length  of  a  Pendulum,—  Let 

C  q  (Fig.  112)  be  the  axis  of  a  pen- 


112 


*»»•  11S- 


dulum  in  which  all  its  weight  is  col- 
lected, C  the  point  of  suspension, 
6*  the  centre  of  gravity,  0  the  cen- 
tre of  oscillation,  a  b,  &c.,  particles 
above  0,  which  accelerate  it,  p,  q, 
&c.,  particles  below  0,  which  retard 
it.  C  0  =  I,  is  the  length  of  the 
pendulum  required.  Denote  the 
masses  concentrated  in  a,  b  .  .  .  .  p,  q, 
by  mt  m' .  .  .  .  m",  m'",  and  their 
distances  from  C  by  r,  r'  .  .  ,  .  r", 
r'"  ;  and  denote  the  distance  from 
C  to  O  by  k.  Denote  the  angular 
velocity,  that  is,  the  velocity  at 
unit's  distance  from  the  centre,  at 
any  instant  by  B  ;  then  the  velocity 
of  m  will  be  r  B  and  its  momentum 
will  be  m  r  B. 

If  m  had  been  placed  at  0,  the 

momentum  would  have  been  ml  B.  The  difference  m  (I  —  r)  0, 
is  that  portion  of  the  force  which  accelerates  the  motion  of  the 
system. 

For  suppose  a  material  particle  m  (Fig.  113)  to  act  upon  a 
pendulum  @  D  without  weight ;  m  at  a  would,  under  the  action 
of  the  component  of  gravity  a  b,  move  the  point  a  to  b  and  swing 
the  pendulum  through  the  angle  B  ;  m  if  transferred  to  o  would, 


-4-  G 


-  -p 


THE    PENDULUM.  103 

gravity  being  the  same,  move  the  point  o  to  x,  and  swing  the 
pendulum  through  an  angle  less  than  6.  Thus  m  at  a  swings  the 
pendulum  through  a  greater  angle  in  a  given  time  than  it  would 
if  at  o,  or  accelerates  the  pendulum,  by  a  force  which  would  carry 
m  over  x  y  in  the  given  time,  or  by  m  (I  —  r)  6  ;  for,  calling 
C  a  =  r  and  C  o  =  I,  ab  =  r  6,  o  x  =  a  b  =r  0,  o  y  =  I  9  ;  then 
o  y  —  ox  =  xy  =  ld  —  r  6  =  (I  —  r)  6,  and  m  moving  with 
velocity  x  y,  or  (I  —  r)  0,  gives  momentum  m  (I  —  r)  B. 

The  moment  of  this  force  with  respect  to  C  is  m  (I  —  r)  r  6. 

In  like  manner  the  moment  of  m'  is  m'  (I  —  r')  r'  0,  and  so  on 
for  all  the  particles  between  C  and  0. 

The  moments  of  the  forces  tending  to  retard  the  system 
applied  at  the  points  p,  q,  &c.,  are 

m"  (r"  -  I)  r"  0,  m"'  (r"f  -  1)  r'"  6,  &c. 
But  since  these  forces  are  to  balance  each  other,  we  have 

m  (I  —  r)  r  6  +  m'  (I  -  r')  r'  d  +  &c.  =  m"  (r"  -  I)  r"  0 
+  m">  (r'"_j)r'"0  +  &c.; 


whence  I  = 


m  r  __  m  r    __  m    r 


"2 


m  r  4-  m'  r'  -f-  m"  r"  +  &c. 

Or  I  =  ~ f ,  where  S  denotes  the  sum  of  all  the  terms  similar 

S  (m  r) 

to  that  which  follows  it. 

The  numerator  of  this  expression  is  called  the  moment  of  inertia 
of  the  body  with  respect  to  the  axis  of  suspension,  and  the  de- 
nominator is  called  the  moment  of  the  mass,  with  respect  to  the 
axis  of  suspension. 

By  the  principle  of  moments  m  r  +  m'  r'  -f-  &c.,  or  8  (m  r)  = 
M  k,  where  M  denotes  the  entire  mass  of  the  pendulum  ;  hence, 

.C  /*v>  *.2\ 

by  substitution,  I  = 


That  is,  the  distance  from  the  axis  of  suspension  to  the  centre 
of  oscillation  is  found  by  dividing  the  moment  of  inertia,  with 
respect  to  that  axis,  by  the  moment  of  the  mass  with  respect  to  the 
same  axis. 

161.  The  Point  of  Suspension  and  the  Centre  of 
Oscillation  Interchangeable. — Let  the  pendulum  now  be  sus- 
pended from  an  axis  passing  through  0,  and  denote  by  I'  the 
distance  from  0  to  the  new  centre  of  oscillation.  The  distances  of 
«,  b  .  .  .  .  p,  q,  from  0,  will  be  I  —  r,  I  —  r',  &c.,  and  the  distance 
G  0  will  be  I  -  Jc. 


104  MECHANICS. 

Hence,  from  the  principle  just  established,  we  have 
_  8 


M  (I  -  k) 

But  from  the  preceding  paragraph   I  =  —  Wy—  >    whence 

JM.  Ic 

8(m  r2)  =  M  Jc  I  ;  and  since  I  is  constant,  8  (m  Z2)  =  8(m  +  m'  + 
m"  +  &c.)  Z2  =  Jf  P,  which  values  substituted  above  give 


,  _          -2lSmr        Mlcl      MP-%MJcl  +  Mkl 


"•y  * .  ^ 

This  last  equation  shows  that  the  centre  of  oscillation  and  the 
point  of  suspension  are  interchangeable  ;  that  is,  if  the  pendulum 
were  suspended  from  0,  it  would  vibrate  in  the  same  time  as  when 
suspended  from  C. 

This  fact  is  taken  advantage  of  in  determining  the  length  of 
the  second's  pendulum  at  any  place.    A  solid  bar  A  B  (Fig.  114), 
is  furnished  with  two  knife-edge  axes  C  and  D,  and  a 
sliding  weight  H.     By  adjusting  this  weight  the  bar  can  FlG-  l14- 
be  made  to  oscillate  in  the  same  time  when  suspended      ^ 
upon  either  axis.     The  distance  between  the  knife-edges 
C  and  D  is  the  length  of  an  equivalent  simple  pendulum, 
and  by  comparing  the  time  of  oscillation  with  that  of  a 
pendulum  beating  seconds,  the  time  of  one  oscillation  of 
this  reversible  pendulum  is  obtained;  from  these  data 
the  length  of  the  second's  pendulum  is  readily  com- 
puted. 

162.  Calculation  of  the  Time  of  Oscillation.— 
Let  the  length  of  the  pendulum  A  B  (Fig.  115)  be 
represented  by  ?,  and  the  height  of  the  arc  of  oscillation 
by  B  D.  Suppose  the  pendulum  to  have  moved  from 

Ctoa;  its  acquired  velocity  will  be  v  =  V%  g  x  D  H. 
(Art.  156).  * 

During  the  succeeding  infinitely  small  interval  of  time  t'  it 
will  describe  the  element  of  its  arc  a  c  with  the  velocity  v ;  hence 

, a  c  _  a  c 


Describe  upon  D  B  a  semi-circumference  ;  m  o  is  the  elemen- 


THE    PENDULUM. 


105 


FIG.  115. 


tary  arc  of  the  semi-circumference  having  the  same  altitude  E  H 
as  the  element  a  c.  As  these  arcs  are  elements  of  the  curves, 
they  may  be  regarded  as  straight 
lines,  and  a  b  c  and  m  n  o  become  tri- 
angles. A  H  a  and  a  I  c  are  similar 
triangles,  their  corresponding  sides 
being  perpendicular  two  and  two, 

, .       a  c      A  a 
and  give  the  equation  —r  =  — j?, 

and  because  the  triangles  m  n  o  and 
I  m  H  are   similar,  for   the  reason 

mo        I  m 

assigned  above, ==  — ^  • 

'no       mH 

Divide   the  first  of  these  equa- 
tions  by  the   second,   and  we  get 
ac  _  A  a  x  m  H 
mo  ~  a  H  x  Im 

But  aH*  =  BE(%A  B-  B  H)  =  £  H  (2  I  -  B  H)  and 
m~E*  =  B  H  x  ED,  whence, 

*  VB  H  x  ED 


ac 
m  o 


I 

^VBE(2l-BH)  ~~  Im  V21  —  BH3 
Im  =  $BDsiudBH=  BD-DH, 


or  since 


ac  — 


=r  x  mo, 


and  this  substituted  in  the  value  of  t',  gives 

" 


mo 


x  DH 


21  x  mo 
2g[il-  (BD-DE)}  ' 


or  dividing  both  terms  by  2  V  I,  we  obtain 

A/T  x  mo 


t'  = 


BD  x  A/? 


BD  -DE 


When  the  amplitude  is  small,  the  double  arc  B  C  not  being  over 

5  degrees,    77-7-,  and  -^-j-  are  very  small,  and  their  difference, 
8  I  «  § 

B  D  —  D  E  .  .   .      .  . 

a- ,  is  smaller  still,  and  may  be  neglected,  giving  as  a 


106  MECHANICS. 

result  t'  =  \/  —  X  jr-wj'  The  time  required  to  describe  OB 
is  the  sum  of  the  times  of  describing  the  elements  of  C  B,  or 
calling  this  time  -,  we  have  -  =  \  /  —  x  -^-y-  x  (sum  of  the 

a  fy  \      g  Jj  U 

elementary  arcs  m  o). 

But  the  sum  of  the  elements  ofDmB  corresponding  to  the 

T>     T\ 

elements  of  C  B  is  the  semicircle  D  m  B  itself,  or  ir  —  —  ;  whence 

/& 

-  =  -  \  /  —  =  time  of  semi-oscillation,  or  calling  t  the  time  of 

*  V    9 
a  complete  oscillation,  we  have 


t  = 

9 

163.  Applications  of  the  Formula.  —  From  the  equation 

t  =  TT  \/  —  9  we  get  I  =  ^—^'    Therefore,  the  length  of  a  pen- 

v    y 

dulum  being  known,  the  time  of  one  vibration  is  found  ;  and  on 
the  other  hand,  if  the  time  of  a  vibration  is  known,  the  length 
of  the  pendulum  is  obtained  from  it. 

From  the  same  formulae,  we  find  that  t  oc  \/I,  or 

TJie  time  in  which  a  pendulum  makes  a  vibration  varies  as  the 
square  root  of  the  length. 

As  t  oc  Vl,  .'.  I  oc  t  2  ;  hence,  if  the  length  of  a  seconds  pendu- 
lum equals  Z,  then  a  pendulum  which  vibrates  once  in  two  seconds 
equals  4  ?,  and  one  which  beats  half  seconds  =  J  I,  &c. 

Again,  by  observing  the  length  of  a  pendulum  which  vibrates 
in  a  given  time,  the  force  of  gravity,  g,  may  be  found.  For,  as 

I  =  ^-y,  .'.  g  =  —£-*    And  if  g  varies,  as  it  does  in  different  lati- 

iT  t 

a  tz 
tudes  and  at  different  altitudes,  then  I  =  ^  cc  g  t2  ;   and  if  the 

time  is  constant  (as,  for  example,  one  second),  then  I  oc  g.    Hence, 
The  length  of  a  pendulum  for  Seating  seconds  varies  as  the  force 
of  gravity. 

Also,  t  oc  f  —  j  ;  that  is,  the  time  of  a  vibration  varies  di- 

U 

rectly  as  the  square  root  of  the  length,  and  inversely  as  the  square 
root  of  the  force  of  gravity. 

Since  the  number,  n,  of  vibrations  in  a  given  time  varies 

inversely  as  the  time  of  one  vibration,  therefore  n  oc  (        ,   and 


THE    PENDULUM. 


107 


g  oc  I  n2.     Hence,  if  the  time  and  the  length  of  a  pendulum  are 
given, 

The  force  of  gravity  varies  as  the  square  of  the  number  of  vibra- 
tions. 

1.  What  is  the  length  of  a  pendulum  to  beat  seconds,  at  the 
place  where  a  body  falls  16^  ft.  in  the  first  second? 

Ans.  39.11  inches,  nearly. 

2.  If  39.11  inches  is  taken  as  the  length  of  the  seconds  pendu- 
lum, how  long  must  a  pendulum  be  to  beat  10  times  in  a  minute? 

Ans.  117-J-feet. 

3.  In  London,  the  length  of  a  seconds  pendulum  is  39.1386 
inches ;  what  velocity  is  acquired  by  a  body  falling  one  second  in 
that  place  ?  Ans.  32.19  feet 

164.  The  Compensation  Pendulum. — This  name  is  given 
to  a  pendulum  which  is  so  constructed  that  its  length  does  not 
vary  by  changes  of  temperature.  As  all  substances  expand  by 
heat  and  contract  by  cold,  therefore  a  pendulum  will  vibrate  more 
slowly  in  warm  than  in  cold  weather.  This  difficulty  is  overcome 
in  several  ways,  but  always  by  employing  two 
substances  whose  rates  of  expansion  and  con- 
traction are  unequal.  One  of  the  most  com- 
mon is  the  gridiron  pendulum,  represented  in 
Fig.  116.  It  consists  of  alternate  rods  of 
steel  and  brass,  connected  by  cross-pieces  at 
top  and  bottom.  The  rate  of  longitudinal 
expansion  and  contraction  of  brass  to  that 
of  steel  is  about  as  100  to  61 ;  so  that  two 
lengths  of  brass  will  increase  and  diminish 
more  than  three  equal  lengths  of  steel.  There- 
fore, while  there  are  three  expansions  of  steel 
downward,  two  upward  expansions  of  brass 
can  be  made  to  neutralize  them.  In  the 
figure  the  dark  rods  represent  steel,  the  white 
ones  brass.  Suppose  the  temperature  to  rise, 
the  two  outer  steel  rods  (acting  as  one)  let 
down  the  cross-bar  d ;  the  two  brass  rods 
standing  on  d  raise  the  bar  b ;  the  steel  rods 
suspended  from  b  let  down  the  bar  e,  on 
which  the  inner  brass  rods  stand,  and  raise  the  short  bar  c ;  and 
finally,  the  centre  steel  rod,  passing  freely  through  d  and  e,  lets 
down  the  disk  of  the  pendulum.  These  lengths  (counting  each 
pair  as  a  single  rod)  are  adjusted  so  as  to  be  in  the  ratio  of  100 
for  the  steel  to  61  for  the  brass ;  in  which  case  the  upward 
expansions  just  equal  those  which  are  downward,  and  therefore 


108  MECHANICS. 

the  centre  of  oscillation  remains  at  the  same  distance  from  the 
point  of  suspension. 

If  the  temperature  falls,  the  two  contractions  of  brass  are  equal 
to  the  three  of  steel,  so  that  the  pendulum  is  not  shortened  by 
cold. 

The  mercurial  pendulum  consists  of  a  steel  rod  terminating  at 
the  bottom  with  a  rectangular  frame  in  which  is  a  tall  narrow  jar 
containing  mercury,  which  is  the  weight  of  the  pendulum.  It 
requires  only  6.31  inches  of  mercury  to  neutralize  the  expansions 
and  contractions  of  42  inches  of  steel.  See  Appendix  for  calcula- 
tions of  the  place  of  the  centre  of  oscillation. 


CHAPTER    VIII. 

CENTRAL    FORCES. 

165.  Central  Forces  Described. — Motion  in  a  curve  is 
always  the  effect  of  two  forces;  one  an  impulse,  which  alone 
would  cause  uniform  motion  in  a  straight  line  ;  the  other  a  con- 
tinued force,  which  urges  the  body  toward  some  point  out  of  the 
original  line  of  motion.  The  first  is  called  the  projectile  force, 
the  second  the  centripetal  force. 

Suppose  a  point  m  (  Fig  117)  to  be  acted  upon  by  an  impulse, 
in  direction  and  intensity  re- 
presented  by  b  m,  and  also  by 
a  constant  force,  m  d.  This 
centripetal  force  m  d  may  be 
resolved  into  two  components; 
one  m  a  in  the  direction  of 
the  tangent,  the  other,  m  h, 
perpendicular  to  it.  The  tan- 
gential component  will  accel- 
erate or  retard  the  motion  in  the  curved  path  according  as  it  acts 
with  the  projectile  force,  or  in  opposition  to  it,  while  the  com- 
ponent at  right  angles  to  this  tends  to  deflect  the  body  from  a 
rectilinear  path,  and  therefore  determines  the  character  of  the 
curve  at  any  instant. 

When  the  body  moves  in  the  circumference  of  a  circle,  the 
tangential  component  of  the  centripetal  force  is  0,  and  hence  the 
motion  is  uniform. 

If  the  centripetal  force  should  cease  to  act  at  any  instant,  the 


CENTRIFUGAL    FORCE. 


109 


body,  by  its  inertia,  would  immediately  begin  to  move  in  a  straight 
line  tangent  to  the  curve  at  the  point  where  the  body  was  when 
the  force  ceased  to  act. 

Since  the  body,  by  its  inertia,  tends  to  move  in  a  tangent, 
there  is  a  continued  resistance  to  deflection  into  a  curved  path, 
equal  and  opposed  to  the  component  m  h,  in  the  direction  of  the 
radius  of  curvature  at  the  instant  ;  this  is  called  the  centrifugal 
force. 

166.  Expressions  for  the  Centrifugal  Force  in  Circu- 
lar Motion.— 

1.  Let  r=the  radius  of  the  circle,  v=. 
fche  velocity  of  the  body,  c  =  the  distance 
through  which  the  centrifugal  force  causes 
the  body  to  move  in  one  second,  and  let 
A  B  (Fig.  118)  be  the  arc  described  in  the 
infinitely  small  time  t  ;  then  A  B  =  v  t, 
and,  by  a  method  similar  to  that  employed 
in  the  discussion  of  the  force  of  gravity,  it 
may  be  shown  that  B  D  —  c  tz. 

But   A    B,   being  a  very    small   arc, 
may  be  considered  as    equal  to  its  chord,   which   is  a    mean 
proportional    between    A   E   and    the    diameter    2  r.      Hence 


FIG.  118. 


,2 
ct*  = 


2r' 


or 


v* 


(1) 


If  this  be  doubled,  then  (Art.  26)  —  is  the  velocity  which  the 

centrifugal  force  is  capable  of  generating  in  one  second,  and  this 
is  sometimes  taken  as  the  measure  of  the  centrifugal  force. 

From  (1)  it  follows,  that  in  equal  circles  the  centrifugal  force 
varies  as  the  square  of  the  velocity. 

2.  The  value  of  c  may  be  expressed  in  a  different  form.    Let 
f  =  the  time  of  a  complete  revolution  ;  then  2  TT  r  =  v  t' ;  whence 

v  =  —77—-    This  substituted  in  (1). gives 

t  • 

2  7T2  r 

Hence,  The  centrifugal  force  varies  directly  as  the  radius  of  the 
circle,  and  inversely  as  the  square  of  the  time  of  revolution. 

3.  Let  w  —  the  weight  of  the  revolving  body,  and  c'  =  the 
centrifugal  force  expressed  in  pounds ;  then 

/  v*       i-  ,     wv*  /ox 

w  :  c  : :  \g  :  —-  :  whence  c  = .     .     (3) 

**    2r'  rg 


110 


MECHANICS. 


FIG.  119. 


Let  n  =  the  number  of  revolutions  per  second ;  then 
v  =  2  TT  r  n,  and  (3)  becomes 

c'  — .  w .  r .  n2    . (4) 

9 

167.  Two  Bodies  Revolving  about  their  Centre  of 
Gravity. — Let  A  and  B  (Fig.  119)  be  two  bodies  connected  by 
a  rod,  and  let  them  be  made  to 

revolve  about  the  centre  of 
gravity  C\  then  by  (4)  the 
centrifugal  force  of  A  will  be 

—  .  A  .  A  C.  n\  and  of  B,  — .  B  .  B  C.  n\ 
9  9 

But  C  being  the  centre  of  gravity  of  the  two  bodies,  A  .  A  C= 
B  .  B  G\  .*.  the  centrifugal  force  of  A  equals  that  of  B.  Hence, 
If  two  bodies  revolve  in  the  same  time  about  an  axis  passing  through 
their  centre  of  gravity,  there  will  be  no  strain  upon  that  axis. 

168.  Centrifugal  Force  on  the  Earth's  Surface.— As  the 

earth  revolves  upon  its  axis,  all  free  particles  upon  it  are  influenced 
by  the  centrifugal  force.  Let  N  8  (Fig.  120)  be  the  axis,  and  A  a 
particle  describing  a  circumference  with  the  radius  A  0.  Put  r= 
CQ,  r'  =  A  0,  I  =  the  angle  A  C  Q, 
the  latitude,  c  =  the  centrifugal  force 
at  the  equator,  c'  —  the  centrifugal 
force  at  A,  v  =  velocity  of  Q,  and  v' 
=  velocity  of  A  ;  then 


FIG   120. 


But  v 


v'  :  :  r 


=_ 
whence  v'  = 


— .     Comparing  the  value  of  c' 


vr' 

— .    Again,  from  the  triangle  A  C  0 

we  have  r'  =  r  cos  I ;   hence  v'  = 

7        -,    ,       v2  cos2 1      v2  cos  I 

v  cos  I,  and  c  =  = .  =  - 

2  r  cos  I          *  •/- 

with  that  of  c,  we  have 

c'  =  c  cos  I. 

That  is,  the  centrifugal  force  at  any  point  on  the  earth's  surface  is 
equal  to  the  centrifugal  force  at  the  equator,  multiplied  by  the  cosine 
of  the  latitude  of  the  place. 

A  body  at  the  equator  loses  by  centrifugal  force  -gfa  part  of 
the  weight  which  it  would  have  if  the  earth  did  not  revolve  on 
its  axis. 

Let  A  J?  represent  the  centrifugal  force  at  A,  and  resolve  it 


CENTRAL    FORCES.  Ill 

into  A  D  on  C  A  produced,  and  A  F,  tangent  to  the  meridian 
N  Q  S ;  then,  since  the  angle  DAB  —  ACQ  =  l,  we  have 

A  D  —  A  B  cos  I  =  c  cos  I .  cos  I  =  c  cos2 1. 
That  is,  that  component  of  the  centrifugal  force  at  any  point,  which 
opposes  the  force  of  gravity,  is  equal  to  the  centrifugal  force  at  the 
equator,  multiplied  ~by  the  square  of  the  cosine  of  the  latitude  of 
the  place. 

In  like  manner  we  find  A  F  =  A  B  sin  I  =  c  cos  I  sin  /  = 

— -^j- — .    From  this  equation  we  see  that  the  tangential  com- 

/& 

ponent  is  0  at  the  equator,  increases  till  I  =  45° ;  where  it  is  a 
maximum  ;  then  goes  on  diminishing  till  I  =  90°,  when  it  again 
becomes  0. 

The  effect  of  A  D  is  to  diminish  the  weight  of  the  particle, 
while  the  effect  of  A  Fis  to  urge  it  toward  the  equator. 

169.  Examples  on  Central   Forces.— 

1.  A  ball  weighing  10  Ibs.  is  whirled  around  in  a  circumference 
of  10  feet  radius,  with  a  velocity  of  30  feet  per  second.     What  is 
the  tension  upon  the  cord  which  restrains  the  ball  ? 

Ans.  28  Ibs.  nearly. 

2.  With  what  velocity  must  a  body  revolve  in  a  circumference 
of  5  feet  radius,  in  order  that  the  centrifugal  force  may  equal  the 
weight  of  the  body  ?  Ans.  v  =  12.7  ft. 

3.  A  ball  weighing  2  Ibs.  is  whirled  round  by  a  sling  3  feet 
long,  making  4  revolutions  per  second.     What  is  its  centrifugal 
force?  Ans.  117.84  Ibs. 

4.  A  weight  of  5  Ibs.  is  attached  to  the  end  of  a  cord  3  feet 
long  just  capable  of  sustaining  a  weight  of  100  Ibs.     How  many 
revolutions  per  second  must  the  body  make  in  order  that  the 
cord  may  be  upon  the  point  of  breaking  ?      Ans.  n=2.3  nearly 

5.  A  railway  carriage,  weighing  7  tons,  moving  at  the  rate  of 
30  miles  per  hour,  describes  an  arc  whose  radius  is  400  yards. 
What  is  the  outward  pressure  upon  the  track?     Ans.  786 -fibs. 

170.  Composition  of  two  Rotary  Motions.— 

When  a  body  is  rotating  on  an  axis,  and  a  force  is  applied 
which  alone  ivould  cause  it  to  rotate  on  Mine  other  axis?  the  body 
will  commence  rotation  on  an  axis  lying  Between  thent,  an*d  the 
velocities  of  rotation  on  the  three  axes  are  such,  that  each  may 
be  represented  by  the  sine  of  the  angle  between  the  other  two. 

Suppose  a  body  is  rotating  on  an  axis  A  B  in  the  plane  of 
H  K,  and  that  a  force  is  applied  to  make  it  rotate  on  the  axis 
C  D  in  the  same  plane  H  K,  these  two  axes  intersecting  within 
the  body  at  some  point  called  G. 


MECHANICS. 


FIG.  121. 


Imagine  a  perpendicular  to  the  plane  of  the  axes  to  be  drawn 
through  G,  and  let  P  be  a  particle  of 
the  body  in  this  perpendicular.  Sup- 
pose the  particle  P,  in  an  infinitely 
small  time  t,  to  pass  over  P  a  perpen- 
dicular to  A  B,  by  the  first  rotation, 
and  over  P  c\  perpendicular  to  C  D,  by 
the  second.  Then,  since  the  particle 
will  describe  the  diagonal  P  e  in  the 
time  t,  this  line  must  indicate  the 
direction  and  velocity  of  the  resultant 
rotation.  Therefore,  if  E  F  be  drawn 
through  G,  perpendicular  to  the  plane 
G  P  e,  E  F  is  the  axis  on  which  the 
body  revolves  in  consequence  of  the  two  rotations  given  to  it. 
Since  P  G  is  perpendicular  to  the  plane  A  G  C,  and  also  to  the 
line  E  F,  therefore  E  Fis  in  that  plane  ;  that  is,  the  new  axis  of 
rotation  is  in  the  plane  of  the  other  two  axes.  The  angles  A  G  Ht 
and  E  G  C,  are  respectively  equal  to  the  angles  a  P  e  and  e  P  c, 
the  inclinations  of  the  planes  of  rotation.  But  the  lines,  P  a, 
P  c,  P  e,  represent  the  velocities  in  those  directions  respectively : 
and  (Art.  44)  P  a  :  P  c  :  P  e  : :  sin  c  P  e  :  sin  a  P  e  :  sin  a  P  c  ; 
therefore  P  a  :  P  c  :  P  e  : :  sin  C  G  E  :  sin  A  G  E  :  sin  A  G  C', 
or,  the  velocities  on  the  three  axes,  (namely,  the  axes  of  the  com- 
ponent rotations,  and  of 

FIG.  122.  the  resultant  rotation,) 

are  such,  that  each  may 
be  represented  by  the 
sine  of  the  angle  between 
the  other  two  axes. 

171.  The  Gyro- 
scope.— The  gyroscope 
affords  an  illustration 
of  the  composition  of 
two  rotations  imparted 
to  a  body.  As  usually 
constructed,  it  consists 
of  a  heavy  wheel  G  IT 
(Fig.  122),  accurately 
balanced  on  the  axis 
a  b,  which  runs  with 
as  little  friction  as  pos- 
sible upon  pivots  in  a  metallic  ring.  In  the  direction  of  the 
axis,  there  is  a  projection  B  from  the  ring,  having  a  socket 


THE    GYROSCOPE.  113 

sunk  into  it  on  the  under  side,  so  that  it  may  rest  on  the  pointed 
standard  S,  without  danger  of  slipping  off. 

The  wheel  is  made  to  rotate  swiftly  by  drawing  off  a  cord 
wound  upon  a  I,  and  then  the  socket  in  B  is  placed  on  the 
standard,  and  the  whole  left  to  itself.  Immediately,  instead  of 
falling,  the  ring  and  wheel  commence  a  slow  revolution  in  a 
horizontal  plane  around  the  standard,  the  point  A  following  the 
circumference  A  E  F,  in  a  direction  contrary  to  the  motion  of  the 
top  of  the  wheel. 

This  revolution  is  explained  by  applying  the  principle  of  com- 
position of  rotations  given  in  the  preceding  article.  The  particles 
of  the  wheel  are  rotating  about  the  horizontal  axis  a  b  by  the  force 
imparted  by  the  string.  The  force  of  gravity  tends  to  make  it 
fall,  that  is,  to  revolve  in  a  vertical  circle  around  the  axis  C  D  at 
right  angles  to  a  b.  Hence,  in  a  moment  after  dropping  the  ring, 
the  system  will  be  found  revolving  on  an  axis  which  lies  in  the 
direction  E  B,  between  A  B  and  G  D,  the  other  two  axes.  Now, 
gravity  bears  it  down  around  a  new  axis  perpendicular  to  E  B. 
Therefore,  as  before,  it  changes  to  still  another  axis  F  B,  and  thus 
continues  to  go  round  in  a  horizontal  circle. 

The  only  way  possible  for  it  to  rotate  on  an  axis  in  a  new  posi- 
tion, is  to  turn  its  present  axis  of  rotation  into  that  position. 
Hence,  the  whole  instrument  turns  about,  in  order  that  its  axis 
may  take  these  successive  positions. 

The  change  of  axis  is  seen  also  by  observing  the  resultant  of 
the  motions  of  the  particles  at  the  top  and  bottom  of  the  wheel. 
For  example,  G  is  moving  swiftly  in  the  direction  m  by  the  rota- 
tion around  a  b ;  by  gravity  it  tends  to  move  slowly  in  the  line  r, 
tangent  to  a  vertical  circle  about  the  centre  B.  The  resultant  is 
in  the  line  n,  tangent  to  the  wheel  when  its  axis  a  1)  has  taken  the 
new  position  E  B. 

The  centre  of  gravity  of  the  ring  and  wheel  tends  to  remain  at 
rest,  while  the  resultant  of  the  two  rotations  carries  around  it  all 
other  parts,  standard  included,  in  horizontal  circles.  But  the 
standard  by  its  inertia  and  friction  resists  this  effort,  and  the  re- 
action causes  the  ring  and  wheel  to  go  around  the  standard.. 

8 


PAET    II, 

HYDROSTATICS. 


CHAPTER    I. 

LIQUIDS    AT    REST. 

172.  Liquids  Distinguished  from  Solids  and  Gases. — 

A.  fluid  is  a  substauce  whose  particles  are  moved  among  each  other 
by  a  very  slight  force.  In  solid  bodies  the  particles  are  held  by 
the  force  of  cohesion  in  fixed  relations  to  each  other ;  hence  such 
bodies  retain  their  form  in  spite  of  gravity  or  other  small  forces 
exerted  upon  them.  If  a  solid  be  reduced  to  the  finest  powder, 
still  each  grain  of  the  powder  is  a  solid  body,  and  its  atoms  are 
held  together  in  a  determinate  shape.  A  pulverized  solid,  if  piled 
up,  will  settle  by  the  force  of  gravity  to  a  certain  inclination,  ac- 
cording to  the  smallness  and  smoothness  of  its  particles,  while  a 
liquid  will  not  rest  till  its  surface  is  horizontal. 

Fluids  are  of  two  kinds,  liquids  and  gases.  In  a  liquid,  there 
is  a  perceptible  cohesion  among  its  particles ;  but  in  a  gas,  the 
particles  mutually  repel  each  other.  These  fluids  are  also  distin- 
guished by  the  fact  that  liquids  cannot  be  compressed  except  in  a 
very  slight  degree,  while  the  gases  are  very  compressible.  A  force 
of  15  pounds  on  a  square  inch,  applied  to  a  mass  of  water,  will 
compress  it  only  about  .000046  of  its  volume,  as  is  shown  by  an 
instrument  devised  by  Oersted.  But  the  same  force  applied  to  a 
quantity  of  air  of  the  usual  density  at  the  earth's  surface  will  re- 
duce it  to  one-half  of  its  former  volume. 

173.  Transmitted  Pressure.— It  is  an  observed  property  of 
fluids  that  a  force  which  is  applied  to  one 

TTV-—          "1  OQ 

part  is  transmitted  undivided  to  all  parts. 
For  instance,  if  a  piston  A  (Fig.  123)  is 
pressed  upon  the  water  in  the  vessel  ADC 
with  a  force  of  one  pound,  every  other  pis- 
ton of  the  same  size,  as  B,  C,  D,  or  E,  re- 
ceives a  pressure  of  one  pound  in  addition 
to  the  previous  pressure  of  the  water  itself. 
Hence  the  whole  amount  of  bursting  pres- 
gnre  exerted  within  the  vessel  by  the  weight 


THE    HYDRAULIC    PRESS.  115 

upon  A  equals  as  many  pounds  as  there  are  portions  of  surface 
equal  to  the  area  of  A.  And  if  the  pressure  is  increased  till  the 
vessel  bursts,  the  fracture  is  as  likely  to  occur  in  some  other  part 
as  in  that  toward  which  the  force  is  directed. 

174.  The  Hydraulic  Press. — An  important  application  of 
the  principle  of  transmitted  pressure  occurs  in  Bramah's  hydraulic 
press,  represented  in  Fig.  124.  The  walls  of  the  cylinder  and 

FIG.  124 


reservoir  are  partly  removed,  to  show  the  interior.  A  is  a  small 
forcing  pump,  worked  by  the  lever  M,  by  which  water  is  raised  in 
the  pipe  a  from  the  reservoir  ff,  and  driven  through  the  tube  K 
into  the  cylinder  B,  where  it  presses  up  the  piston  P,  and  the  iron 
plate  on  the  top  of  it,  against  the  substance  above.  At  each  down- 
ward stroke  of  the  small  piston  p,  a  quantity  of  water  is  trans- 
ferred to  the  cylinder  B,  and  presses  up  the  large  piston  with  a 
force  as  many  times  greater  than  that  exerted  on  the  small  one  as 
the  under  surface  of  P  is  greater  than  that  of  p  (Art  173).  If 
the  diameter  of  p  is  one  inch,  and  that  of  P  is  ten  inches,  then 
any  pressure  on  p  exerts  a  pressure  100  times  as  great  on  P.  The 
lever  M  gives  an  additional  advantage.  If  the  distances  from  the 
fulcrum  to  the  rod  p  and  to  the  hand  are  as  1  :  5,  this  ratio  com- 
pounded with  the  other,  1  :  100,  gives  the  ratio  of  power  at  M  to 


116  HYDROSTATICS. 

the  pressure  at  Q  as  1  :  500 ;  so  that  a  power  of  100  Ibs.  exerts  a 
pressure  of  50000  Ibs. 

This  machine  has  the  special  advantage  of  working  with  a 
small  amount  of  friction.  It  is  used  for  pressing  paper  and 
books,  packing  cotton,  hay,  &c. ;  also  for  testing  the  strength  of 
cables  and  steam-boilers.  It  has  been  sometimes  employed  to  rai^c 
great  weights,  as,  for  instance,  the  tubular  bridge  over  the  Menai 
straits ;  the  two  portions,  after  being  constructed  at  the  water 
level,  were  raised  more  than  100  feet  to  the  top  of  the  piers,  by 
two  hydraulic  presses.  The  weight  of  each  length  lifted  at  once 
was  more  than  1800  tons. 

The  relation  of  power  to  weight  in  the  hydraulic  press  is  in 
accordance  with  the  principle  of  virtual  velocities  (Art.  141). 
For,  while  a  given  quantity  of  water  is  transferred  from  the  smaller 
to  the  larger  cylinder,  the  velocity  of  the  large  piston  is  as  much 
less  than  that  of  the  small  one  as  its  area  is  greater.  But  we  have 
seen  that  the  pressures  are  directly  as  the  areas.  Therefore,  in 
this  as  in  other  machines,  the  intensities  of  the  forces  are  inversely 
as  their  virtual  velocities. 

Ex.  1.  A  press  of  the  same  form  as  in  Fig.  124  has  a  piston 
whose  cross  section  is  one  sq.  ft. ;  the  feed-pump  piston  is  2  sq.  ID. 
cross  section,  and  stroke  6  inches.  The  lever  has  a  short  arm  of 
1  ft.  and  long  arm  of  4  ft.  (measured  from  fulcrum  in  each  case), 
.Find  the  greatest  pressure  that  can  be  produced  by  a  man  who 
exerts  a  force  of  174  Ibs.,  friction  and  difference  of  level  of  the 
liquid  in  the  cylinders  being  disregarded.  Ans.  50112  Ibs. 

Ex.  2.  How  many  strokes  of  the  pump  will  it  take  to  raise  the 
press  piston  one  foot  in  the  last  example  ?  Ans.  144. 

175.  Equilibrium  of  a  Fluid. — In  order  that  a  fluid  may 
be  at  rest, 

,  1.  Tlie  pressures  at  any  one  point  must  be  equal  in  all  direc- 
tions. 

2.  The  surface  must  le  perpendicular  to  the  resultant  of  the 
forces  which  act  upon  it. 

Both  of  these  conditions  result  from  the  mobility  of  the  par- 
ticles. It  is  obvious  that  the  first  must  be  true,  since,  if  any 
particle  were  pressed  more  in  one  direction  than  another,  it  would 
move  in  the  direction  of  the  greater 

force,  and  therefore  not  be  at  rest,      Fia- *~5- 

as  supposed. 

In  order  to  show  the  truth  of  the 
second  condition,  let  mp  (Fig.  125) 
represent  the  resultant  of  the  forces 
which  act  on  the  fluid.  Then,  if  the 


CURVATURE    OF    A    LIQUID    SURFACE. 


117 


surface  is  not  perpendicular  to  m  p,  that  force  may  be  resolved 
into  m  q  perpendicular  to  the  surface,  and  m  f  parallel  to  it. 
The  latter,  m  /,  not  being  opposed,  the  particles  move  in  that 
direction. 

As  gravity  is  the  principal  force  which  acts  on  all  the  par- 
ticles, the  surface  of  a  fluid  at  rest  is  ordinarily  level,  that  is,  per- 
pendicular to  a  vertical  or  plumb  line.  If  the  surface  is  of  small 
extent,  it  is  sensibly  a  plane,  though  it  is  really  curved,  because 
the  vertical  lines,  to  which  it  is  perpendicular,  converge  toward 
the  centre  of  the  earth. 

176.  The  Curvature  of  a  Liquid  Surface.— The  earth 
being  7912  miles  in  diameter,  a  distance  of  100  feet  on  its  surface 
subtends  an  angle  of  about  one  second  at  the  centre,  and  there- 
fore the  levels  of  two  places  100  feet  apart  are  inclined  one  second 
to  each  other. 

The  amount  of  depression  for  moderate  distances  is  found  by 
the  formula,  d  =  f  7/2,  in  which  d  is  the  de- 
pression in  feet,  and  L  the  length  of  arc  in 
miles.  Let  B  E  (Fig.  126)  be  a  small  arc  of 
a  great  circle  on  the  earth  ;  then  C  E  is  the 
depression.  As  B  E  is  small,  its  chord  may 
be  considered  equal  to  the  arc,  and  B  0  equal 
to  the  depression.  But  B  G  :  B  E  : :  B  E  : 

B  A  ;  that  is,  d  :  L  : :  L  :  7912  ;  or  d  =  ~. •• 

i  91/v 

In  order  to  express  d  in  feet,  while  the  other 
lines  are  in  miles,  we  have 

7        Z2  X  52802        L*  x  5280 
d  =  ^-r~ ITTT^  =  — -,— .— —  =  f  L2,  very  nearly. 


7912  x  5280 


7912 


This  gives,  for  one  mile,  d  =  8  inches ;  for  two  miles,  d  =  2 
ft.  8  in.;  and  for  100  miles,  d  =  6667  ft.,  &c.  If  a  canal  is  100 
miles  long,  each  end  is  more  than  a  mile  below  the  tangent  to  the 
surface  of  the  water  at  the  other  end. 

177.  The  Spirit  Level. — Since  the  surface  of  a  liquid  at 
rest  is  level,  any  straight  line  which  is  placed  parallel  to  such  a 
surface  is  also  level.  Leveling  instruments  are  constructed  on 
this  principle.  The  most  accurate  kind  is  the  one  called  the 
spirit  level.  Its  most  essential 

part  is  a  glass  tube,  A  B  (Fig.  Fm  12?        

127),  nearly  filled  with  alcohol 
(because  water  would  be  liable 
to  freeze),  and  hermetically  sealed.     The  tube  having  a  little  con- 
vexity upward  from  end  to  end,  though  so  slight  as  not  to  be 


118 


HYDROSTATICS 


visible,  the  bubble  of  air  moves  to  the  highest  part,  and  changes 
its  place  by  the  least  inclination  of  the  tube.  The  tube  is  so  con- 
nected with  a  straight  bar  of  wood  or  metal,  as  D  C  (Fig.  128),  or 
for  nicer  purposes,  with  FlG  128 

a  telescope,  that  the 
bubble  is  at  the  middle 
M  when  the  bar  or  the 
axis  of  the  telescope  is 
exactly  level.  The  tube 
usually  has  graduation  lines  upon  it  for  adjusting  the  bubble 
accurately  to  the  middle. 

178.  Pressure  as  Depth. — From  the  principle  of  equal 
transmission  of  force  in  a  fluid,  it  follows  that,  if  a  liquid  is  uni- 
formly dense,  its  pressure  on  a  given  area  varies  as  the  perpen- 
dicular depth,  whatever  the  form  or  size  of  the  reservoir.  Let  the 
vessel  A  B  C  D  (Fig.  129),  having  the  form  of  a  right  prism,  be 
filled  with  water,  and  imagine  the  water  to  be  divided  by  horizon- 
tal planes  into  strata  of  equal  thickness.  If  the  density  is  every- 
where the  same,  the  weights  of  these  strata  are  equal.  But  the 
pressure  on  each  stratum  is  the  sum  of  the  weights  of  all  the  strata 
above  it.  Therefore,  in  this  case,  the  pressure  varies  as  the  depth. 


FIG.  129. 
A B 


FIG.  130. 


A  B 


But  let  the  reservoir  (Fig.  130)  contain  water.  The  pressure 
of  the  column  A  B  L  Mia  transmitted  equally  in  every  direction 
(Art.  173).  If  the  area  of  the  section  L  M  is  one  sq.  inch  and 
the  weight  of  the  column  A  L  is  one  pound,  then  every  square 
inch  of  the  side  E  H  will  receive  a  pressure  of  one  pound,  on 
account  of  the  column  A  L,  in  addition  to  the  pressure  it  sus- 
tains from  the  contained  water.  So  also  every  square  inch  of  the 
bottom  D  H  will  sustain  an  added  pressure  of  1  lb.,  and  also  every 
square  inch  of  the  top  ME  will  sustain  an  upward  pressure 
of  1  lb.  That  is  to  say,  the  added  pressure  upon  every  part  of 
the  containing  vessel  L  E  H  D,  whose  area  equals  the  area  of  the 
base  of  the  column  A  L  M  B,  is  equal  to  the  weight  of  that 
column. 

Again,  if  the  base  is  smaller  than  the  top,  as  in  the  vessel 


PRESSURE    AS    DEPTH. 


119 


A  B  E  F  (Fig.  131),  then  the  pressure  on  E  F  equals  only  the 
weight  of  the  column  C  D  E  F.  The  water  in  the  surrounding 
space  A  C  E,  B  D  F,  simply  serves  as  a  vertical  wall  to  balance 
the  lateral  pressures  of  the  central  column. 

If  the  surface  pressed  upon  is  oblique  or  vertical,  then  the 
points  of  it  are  at  unequal  depths  ;  in  this  case,  the  depth  of  the 
area  is  understood  to  be  the  average  depth  of  all  its  parts ;  that  is, 
the  depth  of  its  centre  of  gravity. 

If  the  fluid  were  compressible,  the  lower  strata  would  be  more 
dense  than  the  upper  ones,  and  therefore  the  pressure  would 
increase  at  a  faster  rate  than  the  depth. 

The  following  experiment  will  show  that  the  pressure  of  a 
liquid  upon  a  given  base  is  due  to  the  depth  of  the  liquid  and  is 
independent  of  the  volume.  Bend  a  glass  tube  A  B  C,  as  shown 
in  Fig.  132  (a),  and  attach  a  cup  D  E,  into  which  may  be  screwed 


PIG.  132  (a). 


FIG.  132  (&). 


various  shaped  receivers  M,  N,  0,  &c.  Into  the  cup  D  E  pour 
mercury,  which  will  stand  at  a  level,  say  H  D  E.  Now  screw 
into  the  cup  any  one  of  the  receivers,  as  M,  and  pour  in  water  to 
any  desired  height.  The  mercury  will  be  depressed  in  the  cup 
D  E  by  the  pressure  of  the  water,  and  will  rise  in  A  B  to  some 
point  h.  Now  remove  the  vessel  M  and  substitute  in  succession 
each  of  the  others,  filling  to  the  same  height  as  before.  The  mer- 
cury in  each  case  will  rise  to  the  point  li,  showing  that  the  pres- 
sure upon  the  area  of  mercury  in  the  cup  D  E  is  the  same  in  all 
cases,  for  a  given  height  of  liquid. 

179.  Hydrostatic  Paradox.— To  guard  against  a  possible 
misapprehension  in  this  connection,  the  student  must  be  cau- 
tioned to  distinguish  between  the  pressure  upon  the  bottom  and 
the  weight  of  the  contained  liquid. 

In  the  vessel  A  B  C  D  (Fig.  133),  the  pressure  upon  the  bot- 


120 


HYDROSTATICS. 


D 


Fie 

K  13, 

3. 

J27 

r 

"-."-"•-:-:-: 

\ 

•    : 

Jc 

."1  ;-"-_"_ 
v  :  L".- 

v 

^«~«™ 

-:^<-:-:-:-:-::c- 

JP 


torn  is  equal  to  the  area  of  the  base  D  C  in  square  inches  multi- 
plied by  the  weight  of  a  column  of 
water  of  one  sq.  inch  cross-section  and 
height  E  F,  which  product  is  equal  to 
the  weight  of  a  column  of  cross-section 
D  C  and  height  E  F,  or  the  whole 
volume  m  D  C  n.  But  the  weight  of 
the  contained  water  is  less  than  this, 
as  shown  by  the  figure. 

To  illustrate,  suppose  area  of  base 
D  C  =  12  sq.  inches,  A  D  =  1  inch, 
E  F=  11  inches,  and  x  y  =  1  sq.  inch, 
and  call  the  weight  of  one  cubic  inch  of  water  w.  Then  the 
pressure  upon  the  base  D  C  =  12  x  11  X  w  =  132  w.  The 
weight  of  the  liquid  =  12  x  1  X  w  +  10  x  1  X  w  =  22  w. 

The  downward  pressure  upon  the  base  is,  as  above,  132  w.  The 
upward  pressure  upon  the  upper  base  A  B  is  equal  to  the  area  of 
the  ring  A  B  multiplied  by  the  height  m  A,  or  equal  to  11  x 
10  x  w  =  110  w.  Downward  pressure  minus  upward  pressure 
=  weight. 

132  w  —  110  w  =  22  w,  as  before. 

180.  Amount  of  Pressure  in  Water.— One  cubic  foot  of 
water  weighs  1000  ounces,  or  62.5  pounds  avoirdupois.  There- 
fore, the  pressure  on  one  square  foot,  at  the  depth  of  one  foot,  is 
62.5  pounds.  From  this,  as  the  unit  of  hydrostatic  pressure,  it 
is  easy  to  determine  the  pressures  on  all  surfaces,  at  all  depths ; 
for  it  is  obvious  that,  when  the  depth  is  the  same,  the  pressure 
varies  as  the  surface  pressed  upon  ;  and  it  has  been  shown  that, 
on  a  given  surface,  the  pressure  varies  as  the  depth  of  its  centre 
of  gravity ;  it  therefore  varies  as  the  product  of  the  two.  Let 
p  =  pressure ;  a  =  area  pressed  upon ;  and  d  =  the  depth  of  its 
centre  of  gravity;  then  p  =  a  d  x  62.5. 

Depth. 

1  ft 

10   . 


1G 


Lbs.  per  sq.  ft. 

62.5 

625 

. .1000 


Depth.  Lbs.  per  sq.  ft. 

100ft 6,250 

Imile 330,000 

Smiles 1,650,000 


From  the  above  table  it  may  be  inferred  that  the  pressure  on 
a  square  foot  in  the  deepest  parts  of  the  ocean  must  be  not  far 
from  two  millions  of  pounds  ;  for  the  depth  in  some  places  is 
more  than  five  miles,  and  sea-water  weighs  64.37  pounds,  instead 
of  62.5  pounds.  A  brass  vessel  full  of  air,  containing  only  a  pint, 
and  whose  walls  were  one  inch  thick,  has  been  known  to  be  crushed 
in  by  this  great  pressure,  when  sunk  to  the  bottom  of  the 
ocean. 


HYDROSTATIC    PRESSURE 


Owing  to  the  increase  of  pressure  FIG.  134. 

with  depth,  there  is  great  difficulty  in 
confining  a  high  column  of  water  by 
artificial  structures.  The  strength  of 
banks,  dams,  flood-gates,  and  aqueduct 
pipes,  must  increase  in  the  same  ratio 
as  the  perpendicular  depth  from  the 
surface  of  the  water,  without  regard  to 
its  horizontal  extent. 

181.  Column  of  Water  whose 
Weight  Equals  the  Pressure. — A 

oonvenient  mode  of  conceiving  readily 
of  the  amount  of  pressure  on  an  area, 
in  any  given  circumstances,  is  this  : 
consider  the  area  pressed  upon  to  form 
the  horizontal  base  of  a  hollow  prism ; 
let  the  height  of  the  prism  equal  the 
average  depth  of  the  area ;  and  then 
suppose  it  filled  with  water.  The 
weight  of  this  column  of  water  is  equal 
to  the  pressure.  For  the  contents  of 
the  prism  (whose  base  =  a,  and  its 
height  —  d),  =  a  d ;  and  the  weight 
of  the  same  =  a  d  x  62.5  Ibs. ;  which 
is  the  same  expression  as  was  obtained 
above  for  the  pressure. 

On  the  bottom  of  a  cubical  vessel 
full  of  water,  the  pressure  equals  the 
weight  of  the  water  ;  on  each  side  of  the  same  the  pressure  is  one- 
half  the  weight  of  the  water ;  hence,  on  all  the  five  sides  the 
pressure  is  three  times  the  weight  of  the  water ;  and  if  the  top 
were  closed,  on  which  the  pressure  is  zero,  the  pressure  on  the  six 
sides  is  the  same,  three  times  the  weight  of  the  water. 

182.  Illustrations  of  Hydrostatic  Pressure. — A  vessel 
may  be  formed  so  that  both  its  base  and  height  shall  be  great,  but 
its  cubical  contents  small ;  in  which  case,  a  great  pressure  is  pro- 
duced by  a  small  quantity  of  water.     The  hydrostatic  bellows  is 
an  example.     In  Fig.  134  the  weight  which  can  be  sustained  on 
the  lid  D  1  by  the  column  A  D  is  equal  to  that  of  a  prism  or 
cylinder  of  water,  whose  base  is  D  /,  and  its  height  D  A.    It  is 
immaterial  how  shallow  is  the  stratum  of  water  on  the  base,  or 
how  slender  the  tube  A  D,  if  greater  than  a  capillary  size. 

In  like  manner,  a  cask,  after  being  filled,  may  be  burst  by  an 
additional  pint  of  water  ;  for,  by  screwing  a  long  and  slender  pipe 


122  HYDROSTATICS. 

into  the  top  of  the  cask,  and  filling  it  with  water,  the  pressure  is 
easily  made  greater  than  the  strength  of  the  cask  can  bear. 

183.  Determination  of  Thickness  of  Cylinder. — To 
determine  the  thickness  of  plate  required  in  a  cylindrical  vessel 
that  it  may  sustain  a  given  pressure,  we  assume  that  the  bursting 
results  from  tearing  asunder  the  material  of  the  plate. 

Let  ABE  (Fig.  135)  represent  the  cylindrical  vessel ; 
E  D  C  A  B  a  longitudinal  sec- 
tion through  the  axis;  put 
a  =  A  B  =  length  in  inches, 
2  r  =  C  D  =  internal  diaine- 
ter  in  inches,  e=A  C=D  E— 
thickness  of  plate  in  inches, 
T  =  tenacity  of  the  material 
in  Ibs.  per  sq.  inch;  then 

a  x  e  x   T  =  strength,  or 

resistance  to  tearing  apart,  of  section  BAG.  As  there  are  two 
such  sections  which  resist  the  internal  pressure  the  total  strength 
through  the  section  E  D  C  A  B,  is  2  a  x  e  x  T. 

Call  the  internal  pressure  in  Ibs.  per  sq.  inch  P.  The  total 
bursting  pressure  through  the  section  CD,  acting  upward  and 
downward  to  cause  separation  in  that  plane,  is  equal  to  the  area 
multiplied  by  the  pressure  per  sq.  inch,  or  =  rectangle  2  r  x  a 
X  P.  But  at  the  moment  of  rupture  these  two  must  be  equal, 
therefore 

r  x  P 
2aeT=2rxaxP,  whence  e  = — ^ — which  gives  the 

thickness  when  the  internal  diameter,  the  tenacity  and  the  pres- 
sure are  known.  The  longitudinal  section  through  the  axis  is  the 
weakest  longitudinal  section  that  can  be  taken,  hence  we  need 
consider  no  other. 

To  determine  the  thickness  to  withstand  rupture  through  the 
transverse  section  G  F,  we  have,  area  of  section  of  material 
through  G  F  =  -rr  (r  +  e)2  —  n  r2  =  n  e  (e  +  2  r),  and  the 
tenacity  of  section  =  TT  e  (e  +  2  r)  T. 

The  bursting  pressure  upon  the  plane  through  G  F,  exerted 
upon  the  heads  of  the  cylinder,  =  TT  x  r2  x  P.  These  being 
equal  we  have, 

TT  e  (e  4-  2  r)  T  —  TT  r2  P, 


neglecting  - — ,  which  will  usually  be  a  small  fraction,  we  get 
r  P 


LEVEL    IN    CONNECTED    VESSELS. 


123 


FIG.  136. 


Comparing  this  with  the  previous  result  we  find  that  the 
transverse  section  requires  only  half  the  thickness  of  material,  for 
a  given  pressure,  which  is  required  by  the  longitudinal  section, 
hence  this  section  need  not  be  considered  in  determining  the 
thickness. 

184.  The  Same  Level  in  Connected  Vessels. — In  tubes 
or  reservoirs  which  communicate  with  each  other,  water  will  rest 
only  when  its  surface  is  at  the 
same  level  in  them  all.  If  water 
is  poured  into  D  (Fig.  136),  it 
will  rise  in  the  vertical  tube  B, 
so  as  to  stand  at  the  same  level 
as  in  D.  For,  the  pressure  to- 
ward the  right  on  any  cross- 
section  E  of  the  horizontal  pipe 
m  n  equals  the  product  of  its 
area  by  its  depth  below  D.  So 
the  pressure  on  the  same  section 
towards  the  left  equals  the  pro- 
duct of  its  area  by  its  depth  be- 
low B.  But  these  pressures  are 
equal,  since  the  liquid  is  at  rest. 

Therefore  E  is  at  equal  depths  below  B  and  D ;  in  other  words, 
B  and  D  are  on  the  same  level.  The  same  reasoning  applies  to 
the  irregular  tubes  A  and  C,  and  to  any  others,  of  whatever  form 
or  size. 

Water  conveyed  in  aqueducts,  or  running  in  natural  channels 
in  the  earth,  will  rise  just  as  high  as  the  source,  but  no  higher. 

Artesian  wells  illustrate  the  same  tendency  of  water  to  rise  to 
its  level  in  the  diiferent  branches  of  a  tube.  When  a  deep  boring 
is  made  in  the  earth,  it  may  strike  a  layer  or  channel  of  water 
which  descends  from  elevated  land,  sometimes  very  distant.  The 
pressure  causes  it  to  rise  in  the  tube,  and  often  throws  it  many 
feet  above  the  surface.  Fig.  137  shows  an  artesian  well,  through 

FIG.  137. 


124  HYDROSTATICS. 

which  is  discharged  the  water  that  descends  in  the  porous  stratum 
K  K,  confined  between  the  strata  of  clay  A  B  and  G  D. 

A  tube  driven  to  the  water  bed  anywhere  between  A  or  B  and 
the  lowest  point  in  the  diagram,  might  also  bring  water  to  the 
surface  if  the  flow  below  the  end  of  the  tube  were  sufficiently  ob- 
structed by  friction  ;  hence  an  artesian  well  might  be  successfully 
driven  when  the  inclination  of  the  water  bed  is  wholly  in  one 
direction. 

185.  Centre  of  Pressure. — The  centre  of  pressure  of  any 
surface  immersed  in  water  is  that  point  through  which  passes  the 
resultant  of  all  the  pressures  on  the  surface.     It  is  the  point, 
therefore,  at  which  a  single  force  must  be  applied  in  order  to 
counterbalance  all  the  pressures  exerted  on  the  surface.     If  the 
surface  be  a  plane,  and  horizontal,  the  centre  of  pressure  coincides 
with  the  centre  of  gravity,  because  the  pressures  are  equal  on 
every  part  of  it,  just  as  the  force  of  gravity  is.     But  if  the  plane 
surface  makes  an  angle  with  the  horizon,  the  centre  of  pressure  is 
lower  than  the  centre  of  gravity,  since  the  pressure  increases  with 
the  depth.    For  example,  if  the  vertical  side  of  a  vessel  full  of 
water  is  rectangular,  the  centre  is  one-third  of  the  distance  from 
the  middle  of.the  base  to  the  middle  of  the  upper  side.     If  tri- 
angular, with  its  lower  side  horizontal,  the  centre  of  pressure  is 
one-fourth  of  the  distance  from  the  middle  of  the  base  to  the  ver- 
tex.   If  triangular,  with  the  top  horizontal,  the  centre  of  pressure 
is  half  way  up  on  the  bisecting  line. 

[See  Appendix  for  calculations  of  the  place  of  the  centre  of 
pressure.] 

186.  The  Loss  of  Weight  in  Water. — When  a  body  is 
immersed  in  water,  it  suffers  a  pressure  on  every  side,  which  is 
proportional  to  the  depth.     Opposite  components  of  lateral  pres- 
sures, being  exerted  on  surfaces  at  the  same  depth,  balance  each 
other ;  but  this  cannot  be  true  of  the  vertical  pressures,  since  the 
top  and  bottom  of  the  body  are  at  unequal  depths.     The  upward 
pressure  on  the  bottom  exceeds  the  downward  pressure  on  the 
top ;  and  this  excess  constitutes  the  buoyant  power  of  a  fluid, 
which  causes  a  loss  of  weight. 

A  body  immersed  in  water  loses  weight  equal  to  the  weight  of 
water  displaced. 

For  before  the  body  was  immersed,  the  water  occupying  the 
same  space  was  exactly  supported,  being  pressed  upward  more 
than  downward  by  a  force  equal  to  its  own  weight.  The  weight 
of  the  body,  therefore,  is  diminished  by  this  same  difference  of 
pressures,  that  is.  by  the  weight  of  the  displaced  water. 


EQUILIBRIUM    OF    FLOATING    BODIES.  125 

To  show  this  experimentally,  suspend  a  solid  cylinder  A  (Fig. 
138)  below  a  hollow  cylinder  B,  into  which  it  will  fit  with  great 
nicety  ;  attach  both  to  the 

arm  of  a  balance  and  care-  FlG- 138< 

fully  counterpoise  them ; 
now  pour  water,  or  any 
other  liquid,  into  the  beak- 
er C  until  it  is  full,  and 
the  equilibrium  will  be  de- 
stroyed, the  end  of  the 
beam  D  rising.  Fill  the 
cylinder  B  with  the  same 
liquid,  and  when  it  is  ex- 
actly full,  the  cylinder  A 
will  be  found  to  be  sub- 
merged exactly  to  its  upper  edge,  thus  showing  that  the  buoyancy 
of  the  liquid  in  this  case  is  counteracted  by  a  volume  of  the  same 
liquid  equal  to  the  volume  of  the  submerged  body. 

On  the  supposition  of  the  complete  i  ncompressibility  of  water, 
this  loss  is  the  same  at  all  depths,  because  the  weight  of  displaced 
water  is  the  same.  As  water,  however,  is  slightly  compressible, 
its  buoyant  power  must  increase  a  little  at  great  depths.  Calling 
the  compression  .000046  for  one  atmosphere  (=34  feet  of  water), 
the  bulk  of  water  at  the  depth  of  a  mile  is  reduced  by  about  yj-g-, 
and  its  specific  gravity  increased  in  the  same  ratio ;  so  that, 
possibly,  a  body  might  sink  near  the  surface,  and  float  at  great 
depths  in  the  ocean.  But  this  is  not  probable  in  any  case,  since 
the  same  compressing  force  may  reduce  the  volume  of  the  solid 
as  much  as  that  of  the  water.  And,  furthermore,  the  increase  of 
density  by  increased  depth  is  so  slow,  that  even  if  solids  were 
incompressible,  most  of  those  which  sink  at  all  would  not  find 
their  floating  place  within  the  greatest  depths  of  the  ocean.  For 
example,  a  stone  twice  as  heavy  as  water  must  sink  100  miles 
before  it  could  float. 

187.  Equilibrium  of  Floating  Bodies.— If  the  body  which 
is  immersed  has  the  same  density  as  water,  it  simply  loses  its 
whole  weight,  and  remains  wherever  it  is  placed.  But  if  it  is  less 
dense  than  water,  the  excess  of  upward  pressure  is  more  than  suf- 
ficient to  support  it ;  it  is,  therefore,  raised  to  the  surface,  and 
comes  to  a  state  of  equilibrium  after  partly  emerging.  In  order 
that  a  floating  body  may  have  a  stable  equilibrium,  the  three  fol- 
lowing conditions  must  be  fulfilled  : 

1.  It  displaces  an  amount  of  water  whose  weight  is  equal  to  its 


126 


HYDROSTATICS. 


2.  The  centre  of  gravity  of  the  body  is  in  the  same  vertical  line 
with  that  of  the  displaced  water. 

3.  The  metacenter  is  higher  than  the  centre  of  gravity  of  the 
body. 

The  reason  for  the  first  condition  is  obvious  ;  for  both  the  body 
and  the  water  displaced  by  it  are  sustained  by  the  same  upward 
pressures,  and  therefore  must  be  of  equal  weight. 

FIG.  139. 


That  the  second  is  true,  is  proved  as  follows  :  Let  C  (Fig. 
139,  1)  be  the  centre  of  gravity  of  the  displaced  water,  while  that 
of  the  body  is  at  G.  Now  the  fluid,  previous  to  its  removal,  was 
sustained  by  an  upward  force  equal  to  its  own  weight,  acting 
through  its  centre  of  gravity  (7;  and  the  same  upward  force  now 
acts  upon  the  floating  body  through  the  same  point.  But  the 
body  is  urged  downward  by  gravity  in  the  direction  of  the  vertical 
line  A  G'B.  Were  those  two  forces  exactly  opposite  and  equal, 
they  would  keep  the  body  at  rest ;  but  this  is  the  case  only  when 
the  points  C  and  G  are  in  the  same  vertical  line  ;  in  every  other 
position  of  these  points,  the  two  parallel  forces  tend  to  turn  the 
body  round  on  a  point  between  them. 

188.  The  Metacenter. — To  understand  the  third  condi- 
tion, the  metacenter  must  be  defined.  A  floating  body  assumes 
a  position  such  that  the  line  through  the  centres  of  gravity  of  the 
body  and  of  the  displaced  water  shall  be  vertical ;  now,  regard  this 
line  so  determined  as  fixed  with  respect  to  the  body,  moving  with 
it  to  any  degree  of  inclination  ;  then  move  the  body  so  that 
this  line  shall  make  an  indefinitely  small  angle  with  its  vertical 
position  ;  the  intersection  of  the  line  as  now  placed  with  the 
vertical  through  the  new  centre  of  gravity  of  the  displaced  water 
is  called  the  metacentre.  When  the  centre  of  gravity  of  the  body 
G  is  lower  than  the  metacenter,  as  in  Fig.  139,  2,  the  parallel 
forces,  downward  through  G  and  upward  through  (7,  revolve  the 
body  back  to  its  position  of  equilibrium,  which  is  then  called  a 
stable  equilibrium.  But  if  the  centre  of  gravity  of  the  body  is 


FLOATING    OF    HEAVY    SUBSTANCES.  12? 

higher  than  the  metacenter,  as  in  Fig.  139,  3,  the  rotation  is  in 
the  opposite  direction,  and  the  body  is  upset,  the  equilibrium 
being  unstable.  Once  more,  if  the  centre  of  gravity  of  the  body 
is  at  the  metacenter,  the  body  rests  indifferently  in  any  position, 
as,  for  example,  a  sphere  of  uniform  density.  The  equilibrium 
in  this  case  is  called  neutral. 

If  only  the  first  condition  is  fulfilled,  there  is  no  equilibrium  ; 
if  only  the  first  and  second,  the  equilibrium  is  unstable;  if  all  the 
three,  the  equilibrium  is  stable. 

In  accordance  with  the  third  condition,  it  is  necessary  to  place 
the  heaviest  parts  of  a  ship's  cargo  in  the  bottom  of  the  vessel, 
and  sometimes,  if  the  cargo  consists  of  light  materials,  to  fill  the 
bottom  with  stone  or  iron,  called  ballast,  lest  the  masts  and  rig- 
ging should  raise  the  centre  of  gravity  too  high  for  stability. 
On  the  same  principle,  those  articles  which  are  prepared  for  life- 
preservers,  in  case  of  shipwreck,  should  be  attached  to  the  upper 
part  of  the  body,  that  the  head  may  be  kept  above  water.  The 
danger  arising  from  several  persons  standing  up  in  a  small  boat 
is  quite  apparent ;  for  the  centre  of  gravity  is  elevated,  and  liable 
to  become  higher  than  the  metacenter,  thus  producing  an  un- 
stable equilibrium. 

189.  Floating  in  a  Small    Quantity  of  Water.— As 

pressure  on  a  given  surface  depends  solely  on  the  depth,  and  not 
at  all  on  the  extent  or  quantity  of  water,  it  follows  that  a  body 
will  float  as  freely  in  a  space  slightly  larger  than  itself  as  on  .the 
open  water  of  a  lake.  For  instance,  a  ship  may  be  floated  by  a 
few  hogsheads  of  water  in  a  dock  whose  form  is  adapted  to  it.  In 
such  a  case,  it  cannot  be  literally  true  that  the  displaced  water 
weighs  as  much  as  the  vessel,  when  all  the  water  in  the  dock  may 
not  weigh  a  hundredth  part  as  much.  The  expression  "dis- 
placed water"  means  the  amount  which  would  fill  the  place 
occupied  by  the  immersed  portion  of  the  body.  An  experiment 
illustrative  of  the  above  is,  to  float  a  tumbler  within  another  by 
means  of  a  spoonful  of  water  between. 

190.  Floating  of  Heavy  Substances. — A  body  of  the 
most  dense  material  may  float,  if  it  has  such  a  form  given  it  as  to 
exclude  the  water  from  the  upper  side,  till  the  required  amount 
is  displaced.     Ships  are  built  of  iron,  and  laden  with  substances 
of  greater  specific  gravity  than  water,  and  yet  ride  safely  on  the 
ocean.    A  block  of  any  heavy  material,  as  lead,  may  be  sustained 
by  the  upward  pressure  beneath  it,  provided  the  water  is  excluded 
from  the  upper  side  by  a  tube  fitted  to  it  by  a  water-tight  joint. 

191.  Specific  Gravity. — The  weight  of  a  body  compared 


128  HYDROSTATICS. 

with  the  weight  of  the  same  volume  of  the  standard,  is  called  its 
specific  gravity. 

Distilled  water  at  about  39°  F.,  the  temperature  of  its  greatest 
density,  is  the  standard  for  all  solids  and  liquids,  and  common  air, 
at  32°,  for  gases.  Therefore  the  specific  gravity  of  a  solid  or  a 
liquid  body,  is  the  ratio  of  its  weight  to  the  weight  of  an  equal 
volume  of  water  ;  and  the  specific  gravity  of  an  aeriform  body  is 
the  ratio  of  its  weight  to  the  weight  of  an  equal  volume  of  air. 
Hence,  to  find  the  specific  gravity  of  a  soiid  or  liquid,  divide  its 
weight  by  the  weight  of  the  same  volume  of  water  ;  but  in  the 
case  of  a  gas,  divide  by  the  weight  of  the  same  volume  of  air. 

192.  Methods  of  Finding  Specific  Gravity. — 

1.  For  a  solid  heavier  than  water,  divide  its  weight  by  its  loss 
of  weight  in  water. 

The  reason  for  this  rule  is  obvious.  The  weight  which  a  sub- 
merged body  loses  (Art.  186)  is  equal  to  the  weight  of  the  dis- 
placed water,  which  has,  of  course,  the  same  volume  as  the  body ; 
therefore,  dividing  by  the  loss  is  the  same  as  dividing  by  the  weight 
of  the  same  volume  of  water. 

2.  For  a  solid  lighter  than  water,  divide  its  iveight  by  its  weigh  f 
added  to  the  loss  it  occasions  to  a  heavier  body  previously  balanced 
in  water. 

For,  if  the  light  body  be  attached  to  a  body  heavy  enough  to- 
sink  it,  it  loses  all  its  own  weight,  and  causes  loss  to  the  other 
which  was  previously  balanced.  And  the  whole  loss  equals  the 
weight  of  water  displaced  by  the  light  body.  Hence,  as  before, 
we  in  fact  divide  the  weight  of  the  body  by  the  weight  of  the  same 
volume  of  water. 

If  the  body  whose  specific  gravity  is  required  be  soluble  in 
water,  its  specific  gravity  must  be  determined  with  reference  ta 
some  liquid  which  will  not  dissolve  it,  such  as  alcohol,  turpentine, 
a  saturated  solution  of  the  substance  itself,  &c.,  and  then  the 
specific  gravity  so  obtained  must  be  multiplied  by  the  specific 
gravity  of  the  liquid  used,  as  compared  with  water. 

In  all  the  above  cases  any  air  adhering  to  the  bodies  must  be 
removed  after  immersion. 

3.  For  a  liquid,  find  the  loss  which  a  body  sustains  weighed  in 
the  liquid  and  then  in  water,  and  divide  the  first  loss  by  the 
second. 

For  the  first  loss  equals  the  weight  of  the  displaced  liquid, 
and  the  second  that  of  the  displaced  water ;  and  the  volume  in 
each  case  is  the  same,  namely,  that  of  the  body  weighed  in  them. 

But  the  specific  gravity  of  a  liquid  may  be  more  directly  ob- 
tained by  measuring  equal  volumes  of  it  and  of  water  in  a  flask. 


HYDROMETER,  OR  AREOMETER. 


129 


and  finding  the  weight  of  each.     Then  the  weight  of  the  liquid 
divided  by  that  of  the  water  is  the  specific  gravity  required. 

Flasks  for  the  purpose  are  made  with  carefully  ground  stop- 
pers through  which  is  pierced  a  fine  hole  so  that  in  inserting 
the  stopper  there  may  be  an  overflow  through  the  hole,  after  which 
the  flask  having  been  carefully  wiped  off,  it  is  ready  for  weigh- 
ing. 

193.  The  Hydrometer,  or  Areometer. — In  commerce 
and  the  arts,  the  specific  gravities  of  substances 
are  obtained  in  a  more  direct  and  sufficiently 
accurate  way,  by  instruments  constructed  for 
the  purpose.  The  general  name  for  such  instru- 
ments is  the  hydrometer,  or  areometer.  But 
other  names  are  given  to  such  as  are  limited  to 
particular  uses  ;  as,  for  example,  the  alcoometer 
for  alcohol,  and  the  lactometer  for  milk.  The 
hydrometer,  represented  in  Fig.  140,  consists 
of  a  hollow  ball,  with  a  graduated  stem.  Below 
the  ball  is  a  bulb  containing  mercury,  which 
gives  the  instrument  a  stable  equilibrium  when 
in  an  upright  position.  Since  it  will  descend 
until  it  has  displaced  a  quantity  of  the  fluid 
equal  in  weight  to  itself,  it  will  of  course  sink 
to  a  greater  depth  if  the  fluid  is  lighter.  From  the  depths  to  which 
it  sinks,  therefore,  as  indicated  by  the  graduated  stem,  the  cor- 
responding specific  gravities  are  estimated. 

The  sensibility  of  instruments  of  this 
class  is  increased  by  diminishing  the  diam- 
eter of  the  stem. 

Nicholson's  hydrometer  (Fig.  141)  is  the 
most  useful  of  this  class  of  instruments, 
since  it  may  be  applied  to  finding  the 
specific  gravities  of  solid  as  well  as  liquid 
bodies.  In  addition  to  the  hollow  ball  of 
the  common  hydrometer,  it  is  furnished 
at  the  top  with  a  pan  A  for  receiving 
weights,  and  a  cavity  beneath  for  holding 
the  substance  under  trial.  The  instru- 
ment is  so  adjusted  that  when  1000  grains 
are  placed  in  the  pan,  the  instrument  sinks 
in  distilled  water  at  the  temperature  of 
39£°  F.  to  a  fixed  mark,  0,  on  the  stem. 
Calling  the  weight  of  the  instrument  W, 
the  weight  of  displaced  water  is  JF+1000. 


FIG.  141. 


130 


HYDROSTATICS. 


FIG.  142. 


To  find  the  specific  gravity  of  a  liquid,  place  in  the  pan  such  a 
weight  w  as  will  just  bring  the  mark  to  the  surface.  Then  the 
weight  of  the  liquid  displaced  is  W+  w.  But  its  volume  is  equal 
to  that  of  the  displaced  water.  Therefore  its  specific  gravity  is 

W  +w 
"F  +  1000" 

To  find  the  specific  gravity  of  a  solid,  place  in  the  pan  a  frag- 
ment of  it  weighing  less  than  1000  grains,  and  add  the  weight  w 
required  to  sink  the  mark  to  the  water-level.  Then  the  weight  of 
the  substance  in  air  is  1000  —  w.  Remove  the  substance  to  the 
cavity  at  the  bottom  of  the  instrument,  and  add  to  the  weight  in 
the  pan  a  sufficient  number  of  grains  10'  to  sink  the  mark  to  the 
surface.  Then  w'  is  the  loss  of  weight  in  water ;  therefore, 

1000  —  w  .    ,, 

— -, is  the  specific  gravity  of  the  substance. 

194.  Specific  Gravity  of  Liquids  by  Means  of  Heights. 

— This  method  depends  upon  the  fact  that  the  heights  at  which 
columns  of  liquids  will  be  sustained  by  any  given 
atmospheric  pressure  are  inversely  as  their  specific 
gravi  ties. 

Arrange  two  glass  tubes  A  and  B  (Fig.  142) 
connected  at  the  top  with  a  common  outlet  G,  their 
lower  ends  being  immersed  in  the  liquids  contained 
in  the  beakers  E  and  D.  Exhaust  the  air  by  the 
outlet  C  till  the  liquids  rise  to  any  desired  height, 
say  to  A  and  B,  and  suppose  the  height  of  the 
column  A,  measured  from  the  surface  of  the  liquid 
in  beaker  E  to  be  one-half  that  of  the  column  B 
measured  from  the  surface  in  beaker  D ;  then  the 
specific  gravity  of  liquid  E  is  twice  that  of  liquid  D. 

This  method  gives  only  approximate  results, 
depending  upon  the  fineness  of  division  of  the  scales 
used,  corrected  for  capillarity  of  the  tubes. 

195.  Table  of  Specific  Gravities. — An  accurate   knowl- 
edge of  the  specific  gravities  of  bodies  is  important  for  many  pur- 
poses of  science  and  art,  and  they  have  therefore  been  determined 
with  the  greatest  possible  precision.     The  heaviest  of  all  known 
substances  is  platinum,  whose  specific  gravity,  when  compressed 
by  rolling,  is  22,  water  being  1    ;  and  the  lightest  is  hydrogen, 
whose  specific  gravity  is  —  .073,  common  air  being  1.     Now,  as 
water  is  about  800  times  as  heavy  as  air,  it  is  (800  -i-  .073  — ) 
10.959  times  as  heavy  as  hydrogen.     Therefore  platinum  is  about 
(10.059x22-=)  241,000  times  as  heavy  as  hydrogen.     Between 


TABLE    OF     SPECIFIC     GRAVITIES.  131 

theso  limits,  1  and  241,000,  there  is  a  wide  range  for  the  speciiic 
gravities  of  all  other  substances.  As  a  class,  the  common  metals 
are  the  heaviest  bodies  ;  next  to  these  come  the  metallic  ores  ; 
than  the  precious  gems ;  minerals  in  general,  animal  and  vegetable 
substances,  as  shown  in  the  following  table : 

Metals  (pare),  not  including  the  bases  of  the  alkalies  and 

earths,  from   5 — 22 


Platinum 22.0 

Gold 19.25 

Mercury 13.58 

Lead..  .  11.35 


Copper 8.90 

Steel 7.84 

Iron 7.78 

Tin...  .  7.29 


Silver 10.47    Zinc 7.00 

Metallic  ores,  lighter  than  the  pure  metals,  but  usually  . 
above 4.00 

Precious  gems,  as  the  ruby,  sapphire,  and  diamond 3 — 4 

Minerals,  comprehending  most  stony  bodies 2 — 3 

Liquids,   from   ether  highly  rectified   to  sulphuric  acid 

highly  concentrated | — 2 

Acids  in  general,  heavier  than  water. 

Oils  in  general,  lighter  ;  but  the  oils  of  cloves  and  cinna- 
mon are  heavier  than  water  ;  the  greater  part  lie  between 
.9  and  1 9—1 

Milk 1 . 032 

Alcohol  (perfectly  pure) 797 

"       of  commerce 836 

Proof  spirit ' 923 

vYines  ;  the  specific  gravity  of  the  lighter  wines,  as  Cham- 
pagne and  Burgundy,  is  a  little  less,  and  of  the  heavier 
wines,  as  Malaga,  a  little  greater  than  that  of  water. 

Woods,  cork  being  the  lightest,  and  lignum  vitae  the 
heaviest 24 — 1.34 

196.  Floating". — The  human  body,  when  the  lungs  are  filled 
with  air,  is  lighter  than  water,  and  but  for  the  difficulty  of  keep- 
ing the  lungs  constantly  inflated,  it  would  naturally  float.  With 
a  moderate  degree  of  skill,  therefore,  swimming  becomes  a  very 
easy  process,  especially  in  salt  water.  When,  however,  a  man 
plunges,  as  divers  sometimes  do,  to  a  great  depth,  the  air  in  the 
lungs  becomes  compressed,  and  the  body  does  not  rise  except  by 
muscular  effort.  The  bodies  of  drowned  persons  rise  and  float 
after  a  few  days,  in  consequence  of  the  inflation  occasioned  bj 
putrefaction. 

As  rocks  are  generally  not  much  more  than  twice  as  heavy  as 
water,  nearly  half  their  weight  is  sustained  while  they  are  under 
water ;  hence,  their  weight  seems  to  be  greatly  increased  as  soon 
as  they  are  raised  above  the  surface.  It  is  in  part  owing  to  their 
diminished  weight  that  large  masses  of  rock  are  transported  with 


132  HYDROSTATICS. 

great  facility  by  a  torrent.  While  bathing,  a  person's  limbs  feel 
as  if  they  had  nearly  lost  their  weight,  and  when  he  leaves  the 
water,  they  seem  unusually  heavy. 

197.  To  find  the  Magnitude  of  an  Irregular  Body.— 
It  would  be  a  long  and  difficult  operation  to  find  the  exact  con- 
tents of  an  irregular  mineral  by  direct  measurement.  But  it 
might  be  found  with  facility  and  accuracy  by  weighing  it  in  air, 
and  then  finding  its  loss  of  weight  in  water.  The  loss  is  the 
weight  of  a  mass  of  water  having  the  same  volume.  Now,  as 
1000  ounces  avoirdupois  of  water  measure  1728  cubic  inches,  a 
direct  proportion  will  show  what  is  the  volume  of  the  displaced 
water  ;  that  is,  of  the  mineral  itself. 

198.  Cohesion    and    Adhesion. — What   distinguishes   a 
liquid  from  a  solid  is  not  its  want  of  cohesion  so  much  as  the 
mobility  of  its  particles.     It  is  proved  in  many  ways  that  the  par- 
ticles of  a  liquid  strongly  attract  each  other.     It  is  owing  to  this 
that  water  so  readily  forms  itself  into  drops.     The  same  property 
is  still  more  observable  in  mercury,  which,  when  minutely  divided, 
will  roll  over  surfaces  in  spherical  forms.     When  a  disk  of  almost 
any  substance  is  laid  upon  water,  and  then  raised  gently,  it  lifts  a 
column  of  water  after  it  by  adhesion,  till  at  length  the  edge  of  the 
fluid  begins  to  divide,  and  the  column  is  detached,  not  in  all  parts 
at  once,  but  by  a  successive  rupturing  of  the  lateral  surface.     It 
is  proved  that  the  whole  attraction  of  the  liquid  would  be  far  too 
great  to  be  overcome  by  the  force  applied  to  pull  off  the  disk, 
were  it  not  that  it  is  encountered  by  little  and  little,  at  the  edges 
of  the  column.     But  it  is  the  cohesion  of  the  water  which  is  over- 
come in  this  experiment ;  for  the  upper  lamina  still  adheres  to  the 
disk.     By  a  pair  of  scales  we  find  that  it  requires  the  same  force 
to  draw  off  disks  of  a  given  size,  whatever  the  materials  may  be, 
provided  they  are  wet  when  detached.     This  is  what  might  be 
expected,  since  in  each  case  we  break  the  attraction  between  two 
laminae  of  water.     But  if  we  use  disks  which  are  not  wet  by  the 
liquid,  it  is  not  generally  true  that  those  of  different  material  will 
be  removed  by  the  same  force ;  indicating  that  some  substances 
adhere  to  a  given  liquid  more  strongly  than  others. 

These  molecular  attractions  extend  to  an  exceedingly  small 
distance,  as  is  proved  by  many  facts.  A  lamina  of  water  adheres 
as  strongly  to  the  thinnest  disk  that  can  be  used  as  to  a  thick  one ; 
so,  also,  the  upper  lamina  coheres  with  equal  force  to  the  next 
below  it,  whether  the  layer  be  deep  or  shallow. 

199.  Capillary  Action. — This  name  is  given  to  the  molec- 
ular forces,  adhesion  and  cohesion,  when  they  produce  disturbing 


CAPILLARY    ACTION.  133 

effects  on  the  surface  of  a  liquid,  elevating  it  above  or  depressing 
it  below  the  general  level.  These  effects  are  called  capillary,  be- 
cause most  strikingly  exhibited  in  very  fine  (hair-sized)  tubes. 

The  liquid  loill  be  elevated  in  a  concave  curve,  or  depressed  in  a 
convex  curve,  by  the  side  of  the  solid,  according  as  the  attraction 
of  the  liquid  molecules  for  each  other  is  less  or  greater  than  twice 
the  attraction  between  the  liquid  and  the  solid. 


Case  1st.  Let  H  K  (Fig.  143,  1)  and  L  M  be  a  section  of  the 
vertical  side  of  a  solid,  and  of  the  general  level  of  the  liquid.  The 
particle  A,  where  these  lines  meet,  is  attracted  (so  far  as  this  sec- 
tion is  concerned)  by  all  the  particles  of  an  insensibly  small  quad- 
rant of  the  liquid,  the  resultant  of  which  attractions  is  in  the  line 
A  D,  45°  below  A  M.  It  is  also  attracted  by  all  the  particles  in 
two  quadrants  of  the  solid,  and  the  resultants  are  in  the  directions 
A  B,  45°  above,  and  A  B',  45°  below  L  M. 

Now  suppose  the  force  A  D  to  be  less  than  twice  A  B  or  A  B'. 
Cut  off  C  D  =  A  B ;  then  A  B,  being  opposite  and  equal  to  C  D, 
is  in  equilibrium  with  it.  The  remainder  A  C,  being  less  than 
A  B',  their  resultant  A  E  will  be  directed  toward  the  solid  ;  and 
therefore  the  surface  of  the  liquid,  since  it  must  be  perpendicular 
to  the  resultant  of  forces  acting  on  it  (Art.  175),  takes  the  direc- 
tion represented  ;  that  is,  concave  upward. 

Case  2d.  Let  A  D  (Fig.  143,  2),  the  attraction  of  A  toward 
the  liquid  particles,  be  more  than  tivice  A  B,  the  attraction  toward 
a  quadrant  of  the  solid.  Then,  making  CD  equal  to  A  B,  these 
two  resultants  balance  as  before  ;  and  as  A  C  is  greater  than  A  B', 
the  angle  between  A  C  and  the  resultant  A  E  is  less  than  45°, 
and  A  is  drawn  away  from  the  solid.  Therefore  the  surface, 
being  perpendicular  to  the  resultant  of  the  molecular  forces  acting 
on  it,  is  convex  upward. 

Case  3d.  If  A  D  (Fig.  143,  3)  be  exactly  twice  A  B,  then  CD 
balances  A  B,  and  the  resultant  of  A  C  and  A  B'  is  A  E  in  a  ver- 
tical direction  ;  therefore  the  surface  at  A  is  level,  being  neither 
elevated  nor  depressed. 

Case  1st  occurs  whenever  a  liquid  readily  wets  a  solid,  if 
brought  in  contact  with  it,  as,  for  example,  water  and  olean  glass. 


134 


HYDROSTATICS. 


Case  2d  occurs  when  a  solid  cannot  be  wet  by  a  liquid,  as  glass 
and  mercury.  Case  3d  is  rare,  and  occurs  at  the  limit  between 
the  other  two ;  water  and  steel  afford  as  good  an  example  as  any. 

200.  Capillary  Tubes. — In  fine  tubes  these  molecular  forces 
affect  the  entire  columns  as  well  as  their  edges.    If  the  material 
of  the  tube  can  be  wet  by  a  liquid,  it  will  raise  a  column  of  that 
liquid  above  the  level,  at  the  same  time  making  the  top  of  the 
column  concave.     If  it  is  not  capable  of  being  wet,  the  liquid  is 
depressed,  and  the  top  of  the  column  is  convex.     The  first  case  is 
illustrated  by  glass  and  water  ;  the  second  by  glass  and  mercury. 

The  materials  being  given,  the  distance  by  which  the  liquid  is 
elevated  or  depressed  varies  inversely  as  the  diameter.  Therefore 
the  product  of  the  two  is  constant. 

The  amount  of  elevation  and  depression  depends  on  the 
strength  of  the  molecular  forces,  rather  than  on  the  specific 
gravity  of  the  liquids.  Alcohol,  though  lighter  than  water,  is 
raised  only  half  as  high  in  a  glass  tube. 

201.  Parallel   and    Inclined   Plates.— Between  parallel 
plates  a  liquid  rises  or  falls  half  as  far  as  in  a  tube  of  the  same 
diameter.    This  is  because  the  sustaining  force  acts  only  on  two 
sides  of  each  filament,  while  in  a  tube  it  acts  on  all  sides.     There- 
fore, as  in  tubes  the  height  varies  inversely  as  the  diameter,  so  in 
plates  the  height  varies  inversely  as  the  distance  between  them. 

If  the  plates  are  inclined  to  each  other,  having  their  edge  of 
meeting  perpendicular  to  the  horizon,  the  surface  of  a  liquid  rising 
between  them  assumes  the  form  of  a  hyperbola,  whose  branches 
approach  the  vertical  edge,  and  the  water-level,  as  the  asymptotes 
of  the  curve.  This  results  from  the  law  already  stated,  that  the 
height  varies  inversely  as  the  distance  between  the  plates.  Let 
the  edge  of  meeting,^  £T(Fig.  144), 
be  the  axis  of  ordinates,  and  the 
line  in  which  the  level  surface  of 
the  water  intersects  the  glass,  A  P, 
the  axis  of  abscissas.  Let  B  C, 
D  E,  be  any  ordinates,  and  A  B9 
A  D,  their  abscissas,  and  B  L,  DK, 
the  distances  between  the  plates. 
By  the  law  of  capillarity,  the  heights 
BC,DE,  are  inversely  SisBL,D  K. 
But,  by  the  similar  triangles,  A  B  L, 
ADK,BL,DK,  are  as  A  B,  A  D ; 

therefore,  B  0,  D  E,  are  inversely  as  A  B,  A  D ;  and  this  is  a 
property  of  the  hyperbola  with  reference  to  the  centre  and  asymp- 
totes, that  the  ordinates  are  inversely  as  the  abscissas. 


FIG.  144. 


ILLUSTRATIONS    OF    CAPILLARITY.  135 

202.  Effects  of  Capillarity  on  Floating  Bodies.— Some 

cases  of  apparent  attractions  and  repulsions  between  floating 
bodies  are  caused  by  the-  forms  which  the  liquid  assumes  on  the 
sides  of  the  bodies.  If  two  balls  raise  the  water  about  them,  and 
are  so  near  to  each  other  that  the  concave  surfaces  between  them 
meet  in  one,  they  immediately  approach  each  other  till  they 
touch;  and  then,  if  either  be  moved,  the  other  will  follow  it. 
The  water,  which  is  raised  and  hangs  suspended  between  them, 
draws  them  together. 

Again,  if  each  ball  depresses  the  water  around  it,  they  will  also 
move  to  each  other,  and  be  held  together,  so  soon  as  they  are 
near  enough  for  the  convex  surfaces  to  meet.  In  this  case,  they 
are  not  pulled,  but  pushed  together  by  the  hydrostatic  pressure 
of  the  higher  water  on  the  outside. 

Once  more,  if  one  ball  raises  the  water,  and  the  other  d  'presses 
it,  and  they  are  brought  so  near  each  other  that  the  curves  meet, 
they  immediately  move  apart,  as  if  repelled.  For  now  the  equi- 
librium is  destroyed  in  a  way  just  the  reverse  of  the  preceding 
cases.  The  water  between  the  balls  is  too  high  for  that  which  de- 
presses, and  too  low  for  that  which  raises  the  water,  so  that  the 
former  is  pushed  away,  and  the  latter  is  drawn  away. 

The  first  case,  which  is  by  far  the  most  common,  explains  the 
fact  often  observed,  that  floating  fragments  are  liable  to  be 
gathered  into  clusters ;  for  most  substances  are  capable  of  being 
wet,  and  therefore  they  raise  the  water  about  them. 

203.  Illustrations  of  Capillary  Action.— It  is  by  capil- 
lary action  that  a  part  of  the  water  which  falls  on  the  earth  is 
kept  near  its  surface,  instead  of  sinking  to  the  lowest  depths  of 
the  soil.     This  force  aids  the  ascent  of  sap  in  the  pores  of  plants. 
It  lifts  the  oil  between  the  fibres  of  the  lamp- wick  to  the  place  of 
combustion.     Cloth  rapidly  imbibes  moisture  by  its  numerous 
capillary  spaces,  so  that  it  can  be  used  for  wiping  things  dry.     If 
paper  is  not  sized,  it  also  imbibes  moisture  quickly,  and  can  be 
used  as  blot  tiny -paper ;  but  when  its  pores  are  filled  with  sizing, 
to   fit  it  for   writing,   it  absorbs  moisture  only  in  a  slight  de- 
gree and  the  ink  which  is  applied  to  it  must  dry  by  evapora- 
tion. 

The  great  strength  of  the  capillary  force  is  shown  in  the  effects 
produced  by  the  swelling  of  wood  and  other  substances  when  kept 
wet.  Dry  wooden  wedges,  driven  into  a  groove  cut  around  a 
cylinder  of  stone,  and  then  occasionally  wet,  will  at  length  cause 
it  to  break  asunder.  As  the  pores  between  the  fibres  of  a  rope 
run  around  it  in  spiral  lines,  the  swelling  of  a  rope  caused  by 
keeping  it  wet  will  contract  its  length  with  immense  force. 


136  HYDROSTATICS. 

204.  Questions  in  Hydrostatics.— 

1.  The  diameters  of  the  two  cylinders  of  a  hydraulic  press  are 
one  inch  and  one  foot,  respectively  ;  before  the  piston  descends,  the 
column  of  water  in  the  small  cylinder  is  two  feet  higher  than  the 
bottom  of  the  large  piston.     Suppose  that  by  a  screw  a  force  of 
500  Ibs.  is  applied  to  the  small  piston ;  what  is  the  whole  force 
exerted  on  the  large  piston  at  the  beginning  of  the  stroke  ? 

Am.  72098.17  Ibs. 

2.  A  junk  bottle,  whose  lateral   surface  .contained  50  square 
inches,  being  let  down  into  the  sea  3000  feet,  what  pressure  do 
the  sides  of  the  bottle  sustain,  a  cubic  foot  of  sea  water  weighing 
64.37  Ibs.?  Ans.  67052. 08  + Ibs. 

3.  A  Greenland  whale  sometimes  has  a  surface  of  3600  square 
feet ;  what  pressure  would  he  bear  at  the  depth  of  800  fathoms  ? 

Ans.  1,112,313,600  Ibs. 

4.  A  mill-dam,  running  perpendicularly  across  a  river,  slopes 
at  an  angle  of  30  degrees  with  the  horizon.     The  average  depth 
of  the  stream  is  12  feet,  and  its  breadth  500  )^ards ;  required  the 
amount  of  pressure  on  the  dam  ?  Ans.  13,500,000  Ibs. 

5.  A  mineral  weighs  960  grains  in  air,  and  739  grains  in  water  ; 
what  is  its  specific  gravity?  Ans.  4.344. 

6.  What  are  the  respective  weights  of  two  equal  cubical  masses 
of  gold  and  cork,  each  measuring  2  feet  on  its  linear  edge  ? 

Ans.  The  gold  weighs  9625  Ibs.  =  4.812  tons ;  the  cork 
weighs  120  Ibs. 

7.  A  mass  of  granite  contains  5949  cubic  feet.     The  specific 
gravity  of  a  fragment  of  it  is  found  to  be  2.6 ;  what  does  the  mass 
weigh?  Ans.  483.356  tons. 

8.  An  island  of  ice  rises  30  feet  out  of  water,  and  its  upper  sur- 
face is  a  circular  plane,  containing  fths  of  an  acre.   On  the  supposi- 
tion that  the  mass  is  cylindrical,  required  its  weight,  and  depth  be- 
low the  water,  the  specific  gravity  of  sea-water  being  1.0263,  and 
that  of  ice  .92.     Ans.  Weight,  272048  tons  ;  depth,  259.64  feet. 

9.  Wishing  to  ascertain  the  exact  number  of  cubic  inches  in  a 
very  irregular  fragment  of  stone,  I  ascertained  its  loss  of  weight 
in  water  to  be  5.346  ounces  ;  required  its  volume. 

Ans.  9.238  cubic  inches. 

10.  Hiero,  king  of  Syracuse,  ordered  his  jeweller  to  make  him 
a  crown  of  gold  containing  63  ounces.     The  artist  attempted  a 
fraud  by  substituting  a  certain  portion  of  silver ;  which  being  sus- 
pected, the  king  appointed  Archimedes  to  examine  it.     Archi- 
medes, putting  it  into  water,  found  it  displaced    8.2245  cubic 
inches  of  the  fluid ;  and  having  found  that  the  inch  of  gold  weighs 
10.36  ounces,  and  that  of  silver  5.85  ounces,  lie  discovered  what 


VELOCITY    OF    DISCHARGE.  137 

part  of  the  king's  gold  had  been  purloined  ;  it  is  required  to  re- 
peat the  process.  Ans.  28.8  ounces. 

11.  The  specific  gravity  of  lead  being  11.35;  of  cork,  .24 ;  of 
fir,  .45  ;  how  much  cork  must  be  added  to  60  Ibs.  of  lead,  that 
the  united  mass  may  weigh  as  much  as  an  equal  bulk  of  fir  ? 

Ans.  65.8527  Ibs. 


CHAPTER    II. 

LIQUIDS    IN    MOTION. 

205.  Depth  and  Velocity  of  Discharge.  —  From  an  aper- 
ture which  is  small,  compared  with  the  breadth  of  the  reservoir, 
the  velocity  of  discharge  varies  as  the  square  root  of  the  depth.  For 
the  pressure  on  a  given  area  varies  as  the  depth  (Art.  178).  If  the 
area  is  removed,  this  pressure  is  a  force  which  is  measured  by  the 
momentum  of  the  water  ;  therefore  the  momentum  varies  as  the 
depth  (d).  But  momentum  varies  as  the  mass  (q)  multiplied  by 
the  velocity  (v)  ;  hence  q  v  oc  d.  But  it  is  obvious  that  q  and  v 
vary  alike,  since  the  greater  the  velocity,  the  greater  in  the  same 

ratio  is  the  quantity  discharged.     Therefore,  q2  oc  d,  or  q  QC  d*  ; 

also  v2  QC  d,  or  v  oc  d~*. 

Not  only  does  the  velocity  vary  as  the  square  root  of  the  depth 
of  the  orifice,  but  it  is  equal  to  that  acquired  by  a  body  falling 
through  the  depth. 

Let  h  =  the  height  of  the  liquid  above  the  orifice,  and  &'=:the 
height  of  an  infinitely  thin  layer  at  the  orifice. 

If  this  thin  layer  were  to  fall  through  the  height  li\  under  the 
action  of  its  own  weight  or  pressure,  the  velocity  acquired  would 


be  v'  =  V%gV  (Art.  27). 

Denoting  the  velocity  generated  by  the  pressure  of  the  entire 

column  by  v,  we  have,  since  velocity  oc  V  depth, 
v  :  v'  ::  Vh  :  A///,  or 


/.  v  =  \/2  g  h. 

But  V%ghis  also  the  velocity  acquired  in  falling  through  the 
distance  h  (Art.  27). 

From  an  orifice  16.1  feet  below  the  surface  of  water,  the  veloc- 
ity of  discharge  is  32.2  feet  per  second,  because  this  is  the  velocity 
acquired  in  falling  16.1  feet  ;  and  at  a  depth  four  times  as  great, 


138  HYDROSTATICS. 

that  is,  64.4  feet,  the  velocity  will  only  be  doubled,  that  is,  64.4  feet 
per  second. 

As  the  velocity  of  discharge  at  any  depth  is  equal  to  that  of  a 
body  which  has  fallen  a  distance  equal  to  the  depth,  it  is  theoreti- 
cally immaterial  whether  water  is  taken  upon  a  wheel  from  a  gate 
at  the  same  level,  or  allowed  to  fall  on  the  wheel  from  the  top 
of  the  reservoir.  In  practice,  however,  the  former  is  best,  on 
account  of  the  resistance  which  water  meets  with  in  falling 
through  the  air. 

206.  Descent  of  Surface. — When  water  is  discharged  from 
the  bottom  of  a  cylindric  or  prismatic  vessel,  the  surface  descends 
with  a  uniformly  retarded  motion.     For  the  velocity  with  which 
the  surface  descends  varies  as  the  velocity  of  the  stream,  and 
therefore  as  the  square  root  of  the  depth  (Art.  205).     But  this  is  a 
characteristic  of  uniformly  retarded  motion,   that  the  velocity 
varies  as  the  square  root  of  the  distance  from  the  point  where  the 
motion  terminates,  as  in  the  case  of  a  body  ascending  perpendicu- 
larly from  the  earth. 

The  descent  of  the  surface  of  water  in  a  prismatic  vessel  has 
been  used  for  measuring  time.  The  clepsydra,  or  water-clock 
of  the  Romans,  was  a  time-keeper  of  this  description.  The  grad- 
uation must  increase  upward,  as  the  odd  numbers  1,  3,  5,  7,  &c. ; 
since,  by  the  law  of  this  kind  of  motion,  the  spaces  passed  over  in 
equal  times  are  as  those  numbers. 

If  a  prismatic  vessel  is  kept  full,  it  discharges  twice  as  much 
water  in  the  same  time  as  if  it  is  allowed  to  empty  itself.  For  the 
velocity  in  the  first  instance,  is  uniform  ;  and  in  the  second  it  is 
uniformly  retarded,  till  it  becomes  zero.  We  reason  in  this  case, 
therefore,  as  in  regard  to  bodies  moving  uniformly,  and  with 
motion  uniformly  accelerated  from  rest,  or  uniformly  retarded 
till  it  ceases  (Art.  21),  that  the  former  motion  is  twice  as  great  as 
the  latter. 

207.  Discharge  from  Orifices  in  Different  Situations. 

— Other  circumstances  besides  area  and  depth  of  the  aperture  are 
found  to  have  considerable  influence  on  the  velocity  of  discharge. 
Observations  on  the  directions  of  the  filaments  are  made  by  intro- 
ducing into  the  water  particles  of  some  opaque  substance,  having 
the  same  density  as  water,  whose  movements  are  visible.  From 
such  observations  it  appears  that  the  particles  of  water  descend  in 
vertical  lines,  until  they  arrive  within  three  or  four  inches  of  the 
aperture,  when  they  gradually  turn  in  a  direction  more  or  less 
oblique  toward  the  place  of  discharge.  This  convergence  of  the 
filaments  extends  outside  of  the  vessel,  and  causes  the  stream  to 


DISCHARGE    FROM    ORIFICES. 


139 


diminish  for  a  short  distance,  and  then  increase.  The  smallest 
section  of  the  stream,  called  the  vena  contracta,  is  at  a  distance 
from  the  aperture  varying  from  one-half  of  its  diameter  to  the 
tuhole. 

If  water  is  discharged  through  a  circular  aperture  in  a  thin 
plate  in  the  bottom  of  the  reservoir,  and  at  a  distance  from  the 
sides,  as  in  Fig.  145,  1,  the  filaments  form  the  vena  contracta  at  a 
distance  beyond  the  aperture  equal  to  one-half  of.  its  diameter ;  the 
area  of  the  section  at  the  vena  contracta  is  less  than  two-thirds 
(0.64)  of  the  area  of  the  aperture  ;  this  contraction  also  lessens 
the  theoretical  velocity  by  about  four  per  cent.,  leaving  .96  v  for 
the  final  velocity  ;  combining  these  two  causes,  it  is  found  that  for 
circular  orifices  of  ^  to  6  inches  in  diameter,  with  from  four  to 
20  feet  head  of  water,  the  actual  discharge  is  only  .615  of  the 
theoretical  discharge. 

FIG.  145. 


I 


If  the  reservoir  terminates  in  a  short  pipe  or  ajutage.,  whose 
interior  is  adapted  to  the  curvature  of  the  filaments,  as  far  as  to 
the  vena  contracta,  or  a  little  beyond,  as  in  Fig.  145, 2,  it  is  found 
the  most  favorable  for  free  discharge,  which  in  some  cases  reaches 
0.98  of  the  theoretical  discharge.  The  stream  is  smooth  and  pel- 
lucid like  a  rod  of  glass.  The  most  unfavorable  form  is  that  in 
which  the  ajutage,  instead  of  being  external,  as  in  the  case  just 
described,  projects  inward,  as  in  Fig.  145,  3  ;  the  filaments  in  this 
case  reach  the  aperture,  some  ascending,  others  descending,  and 
therefore  interfere  with  each  other.  Hence  the  stream  is  much 
roughened  in  its  appearance,  and  the  flow  is  only  0.53  of  what  is 
due  to  the  size  of  the  aperture  and  its  depth. 

When  the  aperture  is  through  a  thin  plate,  the  contraction  of 
the  stream  and  the  amount  of  discharge  are  both  modified  by  the 
circumstance  of  being  near  one  or  more  sides  of  the  reservoir. 
There  is  little  or  no  contraction  on  the  side  next  the  wall  of  the 
vessel,  since  the  filaments  have  no  obliquity  on  that  side  ;  and  the 
quantity  is  on  that  account  increased.  The  filaments  from  the 
opposite  side  also  divert  the  stream  a  few  degrees  from  the  per- 
pendicular (Fig.  145,  4). 


140  HYDROSTATICS. 

208.  Friction  in  Pipes. — As  has  just  been  stated,  an  ajutage 
extending  to  or  slightly  beyond  the  vena  contracta,  and  adapted 
to  the  form  of  the  stream,  very  much  increases  the  quantity  dis- 
charged ;  but  beyond  that,  the  longer  the  pipe,  the  more  does  it 
impede  the  discharge  by  friction.     For  a  given  quantity  of  water 
flowing  through  a  pipe  the  resistance  of  friction  increases  with 
the  number  of  points  with  which  the  water  comes  in  contact ; 
that  is,  the  resistance  is  in  proportion  to  the  wetted  surface  ;  for 
every  particle  of  water  in  contact  with  the  interior  surface  of  the 
pipe,  acts  as  a  retarding  force.  •    Now  let  f  be  the  resistance 
of  friction  in  a   pipe   of  unit   diameter,  length   and  velocity ; 
then  the  resistance  in  a  pipe  I  feet  long  and  d  feet  in  diameter 
with  a  unit  of  velocity  will  befell',  but  the  quantity  of  water 
delivered  by  this  pipe  will  be  dz  times  that  delivered  by  the 
former,  in  unit  of  time  with  same  velocity,  since  areas  of  cross- 
sections  are  to  each  other  as  squares  of  their  diameters ;  there- 
fore for  the  same  quantity  of  water  delivered,  the  resistance  of 

friction  in  the  latter  pipe  will  be  •',.     or  i-,  that  is  to  say, 

((  Ct 

the  resistance  of  friction  in  pipes  is  directly  as  their  lengths  and 
inversely  as  their  diameters,  the  velocity  being  constant.  In  order, 
therefore,  to  convey  water  at  a  given  rate  through  a  long  pipe,  it 
is  necessary  either  to  increase  the  head  of  water  or  to  enlarge  the 
pipe,  so  as  to  compensate  for  friction. 

An  aqueduct  should  be  as  straight  as  possible,  not  only  to 
avoid  unnecessary  increase  of  length,  but  because  the  force  of  the 
stream  is  diminished  by  all  changes  of  direction.  If  there  must 
be  change,  it  should  be  a  gradual  curve,  and  not  an  abrupt  turn. 
When  a  pipe  changes  its  direction  by  an  angle,  instead  of  a  curve, 
there  is  a  useless  expenditure  of  force  ;  a  change  of  90°  requires 
that  the  head  of  water  should  be  increased  by  nearly  the  height  due 
to  the  velocity  of  discharge.  For  instance,  if  the  discharge  is  eight 
feet  per  second  (which  is  the  velocity  due  to  one  foot  of  fall),  then 
a  right  angle  in  the  pipe  requires  that  the  head  of  water  should  be 
increased  by  nearly  one  foot,  in  order  to  maintain  that  velocity. 

Empirical  formulae,  based  upon  the  results  of  experiments  for 
the  velocity  of  flow  in  pipes,  and  for  the  loss  of  head  due  to  bends 
and  angles  in  the  pipe,  are  given  in  works  treating  of  Practical 
Hydraulics.  The  derivation  and  development  of  such  formulae  is 
beyond  the  scope  of  a  work  like  this. 

209.  Jets. — Since  a  body,  when  projected  upward  with  a  cer- 
tain velocity,  will  rise  to  the  same  height  as  that  from  which  it 
must  have  fallen  to  acquire  that  velocity,  therefore,  if  water  issue 
from  the  side  of  a  vessel  through  a  pipe  bent  upward,  it  would, 


RIVERS. 


141 


were  it  not  for  the  resistance  of  the  air  and  friction  at  the  orifice, 
rise  to  the  level  of  the  water  in  the  reservoir.  If  water  is  dis- 
charged from  an  orifice  in  any  other  than  a  vertical  direction,  it 
describes  a  parabola,  since  each  particle  may  be  regarded  as  a  pro- 
jectile (Art.  47). 

If  a  semicircle  be  described  on  the  perpendicular  side  of  a 
vessel  as  a  diameter,  and  water 
issue  horizontally  from  any  point, 
its  range,  measured  on  the  level  of 
the  base,  equals  tivice  the  ordinate 
of  that  point.  For,  the  velocity 
with  which  the  fluid  issues  from 
the  vessel,  being  that  which  is  due 
to  the  height  B  G  (Fig.  146),  is 

^YgTB~Q  (Art  27).  But  after 
leaving  the  orifice,  it  arrives  at  the 
horizontal  plane  in  the  time  in  which  a  body  would  fall  freely 


through  GD,  which  is 


Since  the  horizontal  motion 


is  uniform,  the  space  equals  the  product  of  the  time  by  the  veloc- 


ity  ;  that  is,  D  E  = 


'=2V£  G.GD= 


%  G  II,  or  twice  the  ordinate  of  the  semicircle  at  the  place  of  dis- 
charge. 

The  greatest  range  occurs  when  the  fluid  issues  from  the 
centre,  for  then  the  ordinate  is  greatest ;  and  the  range  at  equal 
distances  above  and  below  the  centre  is  the  same. 

The  remarks  already  made  respecting  pipes  apply  to  those 
which  convey  water  to  the  jets  of  fire-engines  and  fountains.  If 
the  pipe  or  hose  is  very  long,  or  narrow,  or  crooked,  or  if  the  jet- 
pipe  is  not  smoothly  tapered  from  the  full  diameter  of  the  hose  to 
the  aperture,  much  force  is  lost  by  friction  and  other  resistances, 
especially  in  great  velocities.  If  the  length  of  hose  is  even  twenty 
times  as  great  as  its  diameter,  32  per  cent,  of  height  is  lost  in  the 
jet,  and  more  still  when  the  ratio  of  length  to  diameter  is  greater 
than  this. 

310.  Rivers, — Friction  and  change  of  direction  have  great 
iiiiiii  11  jo  on  the  flow  of  rivers.  A  dynamical  equilibrium,  as  it  is 
called,  exists  between  gravity,  which  causes  the  descent,  and  the 
resistances,  which  prevent  acceleration  at  any  given  point,  beyond 
a  certain  moderate  limit ;  as  the  same  quantity  of  water  must  pass 
every  cross  section  of  the  stream  in  the  same  unit  of  time,  under 
ordinary  conditions,  the  velocity  varies  inversely  as  the  area  of 


143 


HYDROSTATICS. 


the  cross  section.  The  velocity  in  all  parts  of  the  same  section, 
however,  is  not  the  same  ;  it  is  greatest  at  that  part  of  the  surface 
where  the  depth  is  greatest,  and  least  in  contact  with  the  bed  of 
the  stream. 

To  find  the  mean  velocity  through  a  given  section,  it  is  neces- 
sary to  float  bodies  at  various  places  on  the  surface,  and  also  below 
it,  to  the  bottom,  and  to  divide  the  sum  of  all  the  velocities  thus 
obtained,  by  the  number  of  observations.  To  obtain  the  quantity 
of  water  which  flows  through  a  given  section  of  a  river,  having 
determined  the  velocity  as  above,  find  next  the  area  of  the  section, 
by  taking  the  depth  at  various  points  of  it,  and  multiplying  the 
mean  depth  by  the  breadth.  The  quantity  of  water  is  then  found 
by  multiplying  the  area  by  the  velocity. 

The  increased  velocity  of  a  stream  during  a  freshet,  while  the 
stream  is  confined  within  its  banks,  exhibits  something  of  the  ac- 
celeration which  belongs  to  bodies  descending  on  an  inclined 
plane.  It  presents  the  case  of  a  river  flowing  upon  the  top  of 
another  river,  and  consequently  meeting  with  much  loss  resistance 
than  when  it  runs  upon  the  rough  surface  of  the  earth  itself.  The 
augmented  force  of  a  stream  in  a  freshet  arises  from  the  simulta- 
neous increase  of  the  quantity  of  water  and  the  velocity.  In  con- 
sequence of  the  friction  of  the  banks  and  beds  of  rivers,  and  the 
numerous  obstacles  they  meet  with  in  their  winding  course,  their 
velocity  is  usually  very  small,  not  more  than  three  or  four  miles 
per  hour;  whereas,  were  it  not  for  these 
impediments,  it  would  become  immensely 
great,  and  its  effects  would  be  exceedingly 
disastrous.  A  very  slight  declivity  is  suffi- 
cient for  giving  the  running  motion  to 
water.  The  largest  rivers  in  the  world 
fall  about  five  or  six  inches  in  a  mile. 

211.  Hydraulic  Pumps.— The  most 
common  pumps  for  raising  water  operate 
on  a  principle  of  pneumatics,  and  will  be  de- 
scribed under  that  subject. 

In  the  lifting  pump  the  water  is  pushed 
up  in  the  pump  tube  by  a  piston  placed  be- 
low the  water-level.  In  the  tube  A  B  (Fig. 
147)  is  a  fixed  valve  F,  a  little  below  the 
water-level  L  L,  while  still  lower  is  the  pis- 
ton Pt  in  which  there  is  a  valve.  Both  of 
these  valves  open  upward.  The  piston  is 
attached  to  a  rod,  which  extends  downward 
to  the  frame  F  F.  This  frame  can  be  moved 


FIG.  147. 


CENTRIFUGAL    PUMPS.  14? 

up  and  down  on  the  outside  of  the  tube  by  a  lever.  When  the 
piston  descends,  the  water  passes  through  its  valve  by  hydrostatic 
pressure ;  and  when  raised,  it  pushes  the  water  before  it  through 
the  fixed  valve,  which  then  prevents  its  return.  In  this  manner, 
by  repeated  strokes,  the  water  can  be  driven  to  any  height  which 
the  instrument  can  bear. 

The  chain  pump  consists  of  an  endless  chain  with  circular 
disks  attached  to  it  at  intervals  of  a  few  inches,  which  raise  the 
water  before  them  in  a  tube,  by  means  of  a  wheel  over  which  the 
chain  passes;  the  wheel  may  be  turned  by  a  crank.  The  disks 
cannot  fit  closely  in  the  tube  without  causing  too  great  resistance; 
hence,  a  certain  velocity  is  requisite  in  order  to  raise  water  to  the 
place  of  discharge  ;  and  after  the  working  of  the  pump  ceases,  the 
water  soon  descends  to  the  level  in  the  well. 

21 2.  Centrifugal     Pumps.— Water    may    also    be    raised 
through  small  heights  and  in  great  volume  by  the  centrifugal 
pump.     This  consists  of  revolving  curved,  hollow  arms,  connected 
with  a  hollow  axis  through  which  the  water  enters.     As  this  axis 
is  made  to  rotate  in  a  direction  contrary  to  the  curvature  of  the 
arms  the  centrifugal  force  causes  the  water  to  leave  the  arms  and 
move  off  in  tangents  ;  a  casing  drum  inclosing  the  revolving  por- 
tion forces  the  water  to  move  around  in  a  vortex  till  it  reaches  a 
delivery  pipe  entering  the  drum  as  a  tangent,  through  which  it  is 
discharged.     A  high  delivery  requires  so  great  velocity  that  the 
pump  becomes  inferior  in  efficiency  to  other  forms. 

213.  The  Hydraulic    Ram. — When   a  large  quantity  of 
water  is  descending  through  an  inclined  pipe,  if  the  lower  extrem- 
ity is  suddenly  closed,  since  water  is  nearly  incompressible,  the 
shock  of  the  whole  column  is  received  in  a  single  instant,  and  if 
no  escape  is  provided,  is  very  likely  to  burst  the  pipe.     The  inten- 
sity of  the  shock  of  water  when  stopped  is  made  the  means  of 
raising  a  portion  of  it  above  the  level  of  the  head.     The  instru- 
ment for  effecting  this  is  called  the  hydraulic  ram.     At  the  lower 
end  of  a  long  pipe,  P  (Fig.  148),  is  a  valve,  F,  opening  downward  ; 

FIG.  148. 


T 


14* 


HYDROSTATICS. 


FIG.  149. 


near  it,  another  valve,  V,  opens  into  the  air-vessel,  A  ;  and  from 
this  ascends  the  pipe,  T,  in  which  the  water  is  to  be  raised.  As 
the  valve  Flies  open  by  its  weight,  the  water  runs  out,  till  its 
momentum  at  length  shuts  it,  and  the  entire  column  is  suddenly 
stopped ;  this  impulse  forces  the  water  into  the  air-vessel,  and 
thence,  by  the  compressed  air,  up  the  tube  T.  As  soon  as  the 
momentum  is  expended,  the  valve  V  drops,  and  the  process  is  re- 
peated. 

214.  Water-Wheels  with  a  Horizontal  Axis.— The 
overshot  wheel  (Fig.  149)  is  con- 
structed with  buckets  on  the 
circumference,  which  receive 
the  water  just  after  passing  the 
highest  point,  and  empty  them- 
selves before  reaching  the  bot- 
tom. The  weight  of  the  water, 
as  it  is  all  on  one  side  of  a  ver- 
tical diameter,  causes  the  wheel 
to  revolve.  It  is  usually  made 
as  large  as  the  fall  will  allow, 
and  will  carry  machinery  with 
a  very  small  supply  of  water, 
if  the  fall  is  only  considerable. 
The  moment  of  each  bucket-full 
constantly  increases  from  a,  where  it" is  filled,  to  F9  where  its  act- 
ing distance  is  radius,  and  therefore  a  maximum.  From  F  down- 
ward the  moment  decreases,  both  by  loss 
of  water  and  diminution  of  acting  dis- 
tance, and  becomes  zero  at  L.  These 
wheels  deliver  from  70  to  80  per  cent  of 
the  horse-power  of  the  fall  of  water  re- 
ceived upon  them. 

The  undershot  wheel  (Fig.  150)  is 
revolved  by  the  momentum  of  running 
water,  which  strikes  the  float-boards  on 
the  lower  side.  When  these  are  placed, 
as  in  the  figure,  perpendicular  to  the  circumference,  the  wheel 
may  turn  either  way;  this  is  the  construction  adopted  in  tide- 
mills.  When  the  wheel  is  required  to  turn  only  in  one  direction, 
an  advantage  is  gained  by  placing  the  float-boards  so  as  to  present 
an  acute  angle  toward  the  current,  by  which  means  the  water  acts 
partly  by  its  weight,  as  in  the  overshot  wheel.  The  undershot 
wheel  is  adapted  to  situations  where  the  supply  of  water  is  always 
abundant. 


FIG.  150. 


THE    TURBINE. 


145 


The  maximum  efficiency  of  these  wheels  is  obtained  when  the 
circumferential  velocity  is  one  half  the  velocity  of  the  water, 
and  is  about  30  per  cent,  of  the  theoretical  work  of  the  water 
used.  With  curved  float-boards  the  efficiency  may  reach  about 
60  per  cent. 

In  the  breast  wheel  (Fig.  151)  the  water  is  received  upon  the 
float-boards  at  about  the  height  of  the  axis,  and  acts  partly  by  its 
weight,  and  partly  by  its 

momentum.     The  planes  of  FIG.  151. 

the  float-boards  are  set  at 
right  angles  to  the  circum- 
ference of  the  wheel,  and 
are  brought  so  near  the  mill- 
course  that  the  water  is  held 
and  acts  by  its  weight,  as  in 
buckets.  The  efficiency  is 
about  40  to  50  per  cent. 

215.  The  Turbine.— 

This  very  efficient  water- 
wheel,  frequently  called  the 
French  turbine,  is  of  modern 

invention,  and  has  received  its  chief  improvements  in  this  coun- 
try. It  revolves  on  a  vertical  axis,  and  surrounds  the  bottom  of 
the  reservoir  from  which  it  receives  the  water.  The  lower  part 
of  the  reservoir  is  divided  into  a  large  number  of  sluices  by  curved 
partitions,  which  direct  the 
water  nearly  into  the  line  of  a 
tangent,  as  it  issues  upon  the 
wheel.  The  vanes  of  the  wheel 
are  curved  in  the  opposite  di- 
rection, so  as  to  receive  the 
force  of  the  issuing  streams  at 
right  angles.  The  horizontal 
section  (Fig.  152)  shows  the 
lower  part  of  the  reservoir  with 
its  curved  guides,  #,  a,  a,  and 
the  wheel  with  its  curved  vanes, 
v,  v,  v,  surrounding  the  reser- 
voir ;  D  is  the  central  tube, 
through  which  the  axis  of  the 

wheel  passes.  Fig.  153  is  a  vertical  section  of  the  turbine  ;  but 
it  does  not  present  the  guides  of  the  reservoir,  nor  the  vanes  of 
the  wheel.  C  G,  C  G,  is  the  outer  wall  of  the  reservoir  ;  D,  Z>, 
its  inner  wall  or  tube  ;  F,  F,  the  base,  curved  so  as  to  turn  the 


FIG.  152. 


146 


HYDROSTATICS. 


descending  water  gradually  into  a  horizontal  direction.      The 
outer  wall,  which  terminates  at  6r,  G,  is  connected  with  the  base 


and  tube  by  the  guides  which  are  shown  at  a,  a,  in  Fig.  152. 
The  lower  rim  of  the  wheel,  H,  H,  is  connected  with  the  upper 
rim,  P,  P,  by  the  vanes  between  them,  v,  v  (Fig.  152),  and  to 
the  axis,  E,  E,  by  the  spokes  7,  /.  The  gate,  J,  J,  is  a  thin 
cylinder  which  is  raised  or  lowered  between  the  wheel  and  the 
sluices  of  the  reservoir.  The  bottom  of  the  axis  revolves  in  the 
socket  K,  and  the  top  connects  with  the  machinery.  As  the 
reservoir  cannot  be  supported  from  below,  it  is  suspended  by 
flanges  on  the  masonry  of  the  wheel-pit,  or  on  pillars  outside  of 
the  wheel.  To  prevent  confusion  in  the  figure,  the  supports  of 
the  reservoir  and  the  machinery  for  raising  the  gate  are  omitted. 
By  the  curved  base  and  guides  of  the  reservoir,  the  water  is  con- 
ducted in  a  spiral  course  to  the  wheel,  with  no  sudden  change  of 
direction,  and  thus  loses  very  little  of  its  force.  The  wheel  usually 
runs  below  the  level  of  the  water  in  the  wheel-pit,  as  represented 
in  the  figure,  L  L  being  the  surface  of  the  water.  The  reservoir 
is  sometimes  merely  the  extremity  of  a  large  tapering  tube  or 
supply  pipe,  bent  from  a  horizontal  to  a  vertical  direction.  In 
such  a  case,  the  tube  D  D,  in  which  the  axis  runs,  passes  through 
the  upper  side  of  the  supply  pipe.  The  figure  represents  only 
the  lower  part.  The  efficiency  is  about  80  per  cent.,  though 
many  claim  a  much  higher  efficiency  than  this. 


BARKER'S    MILL. 


147 


FIG.  154. 


216.  Barker's  Mill. — This  machine  operates  on  the  prin- 
ciple of  unbalanced  hydrostatic  pressure.     It  consists  of  a  vertical 
hollow  cylinder,  A  B  (Fig.  154),  free  to  revolve  on  its  axis  M  N, 
and  having  a  horizontal  tube  connected 

with  it  at  the  bottom.  Near  each  end 
of  the  horizontal  tube,  at  P  and  P',  is 
an  orifice,  one  on  one  side,  and  one  on 
the  opposite.  The  cylinder,  being  kept 
full  of  water,  whirls  in  a  direction  op- 
posite to  that  of  the  discharging  streams 
from  P  and  P'.  This  is  owing  to  the 
fact  that  hydrostatic  pressure  is  re- 
moved from  the  apertures,  while  on  the 

interior  of  the  tube,  at  points  exactly  j 

opposite  to  them,  are  pressures  which 
are  now  unbalanced,  but  which  would 
be  counteracted  by  the  pressures  at  the 
apertures,  if  they  were  closed.  The 
tube  P  P'  may  revolve  either  in  the 
air,  or  beneath  the  surface  of  the  water. 
The  speed  of  rotation  is  increased  by  lengthening  the  tube  A  B. 

217.  Resistance  to  Motion  in  a  Liquid. — The  resistance 
which  a  body  encounters  in  moving  through  any  fluid  arises  from 
the  inertia  of  the  particles  of  the  fluid,  their  want  of  perfect  mo- 
bility among  each  other,  and  friction.     Only  the  first  of  these 
admits  of  theoretical  determination.     So  far  as  the  inertia  of  the 
fluid  is  concerned,  the  resistance  which  a  surface  meets  with  in 
moving  perpendicularly  through  it  varies  as  the  square  of  the 
velocity.     For  the  resistance  is  measured  by  the  momentum  im- 
parted by  the  moving  body  to  the  fluid.    And  this  momentum  (m) 
varies  as  the  product  of  the  quantity  of  fluid  set  in  motion  (q), 
and  its  velocity  (v)  ;   or  m  oc  q  v.    But  it  is  obvious  that  the 
quantity  displaced  varies  as  the  velocity  of  the  body,  or  q  oc  v ; 
hence  m  oc  v2.     Therefore  the  resistance  varies  as  the  square  of 
the  velocity. 

This  proposition  is  found  to  hold  good  in  practice,  where  the 
velocity  is  small,  as  the  motions  of  boats  or  ships  in  water ;  but 
when  the  velocity  becomes  very  great,  as  that  of  a  cannon  ball, 
the  resistance  increases  in  a  much  higher  ratio  than  as  the  square 
of  the  velocity.  Since  action  and  reaction  are  equal,  it  makes  no 
difference,  in  the  foregoing  proposition,  whether  we  consider  the 
body  in  motion  and  the  fluid  at  rest,  or  the  fluid  in  motion  and 
striking  against  the  body  at  rest. 

Since  the  resistance  increases  so  rapidly,  there  is  a  wasteful 


148 


HYDROSTATICS. 


expenditure  of  force  in  trying  to  attain  great  velocities  in  naviga- 
tion. For,  in  order  to  double  the  velocity  of  a  steamboat,  the 
force  of  the  steam  must  be  increased  four  fold ;  and,  in  order 
to  triple  its  velocity,  the  force  must  become  nine  times  as 
great. 

When  the  resistance  becomes  equal  to  the  moving  force,  the 
body  moves  uniformly,  and  is  said  to  be  in  a  state  of  'dynamical 
equilibrium.  Thus,  a  body  falling  freely  through  the  air  by 
gravity  does  not  continue  to  be  accelerated  beyond  a  certain  limit, 
but  is  finally  brought,  by  the  resistance  of  the  air,  to  a  uniform 
motion. 

218.  Water  Waves. — These  are  moving  elevations  of  water, 
caused  by  some  force  which  acts  unequally  on  its  surface.     There 
are  two  very  different  kinds  of  waves,  called,  respectively,  ivaves 
of  oscillation  and  -waves  of  translation.     In  the  first  kind  the  par- 
ticles of  water  have  a  vibratory  or  reciprocating  motion,  by  which 
the  vertical  columns  are  alternately  lengthened  and  shortened.    A 
familiar  example  of  this  kind  is  the  sea-wave.    In  the  waves  of 
translation  the  particles  are  raised,  transferred  forward,  and  then 
deposited  in  a  new  place,  without  any  vibratory  movement. 

219.  Waves  of  Oscillation. — If  a  pebble  be  tossed  upon 
still  water,  it  crowds  aside  the  par  tides  beneath  it,  and  raises  them 
above  the  level,  forming  a  wave  around  it  in  the  shape  of  a  ring. 
As  soon  as  this  ring  begins  to  descend,  it  elevates  above  the  level 
another  portion  around  itself,  and  thus  the  ring- wave  continues 
to  spread  outward  every  way  from  the  centre.     But  in  the  mean- 
time the  water  at  the  centre,  as  it  rises  toward  the  level,  acquires 
a  momentum  which  lifts  it  above  that  level.     From  that  position 
it  descends,  and  once  more  passes  below  the  level,  thus  starting  a 
new  wave  around  it,  as  at  first,  only  of  less  height.    Hence,  we  see 
as  the  result  of  the  first  disturbance,  a  series  of  concentric  waves 
continually  spreading  outward  and  di- 
minishing in    height    at   greater   dis- 
tances, until  they  cease   to  be  visible. 

In  Fig.  155  are  represented  three  circu- 
lar waves  at  one  of  the  moments  of 
time  when  the  ceriter  is  lowest.  The 
shaded  parts  are  the  basins  or  troughs, 
and  the  light  parts,  c,  c,  c,  are  the  ridges 
or  crests.  Fig.  156  is  a  vertical  section 
along  the  line,  c,  c,  through  the  centre 
of  the  system,  corresponding  to  the  mo- 
mentary arrangement  of  Fig.  155.  The  central  basin  is  at  I,  and 


FIG.  1F5. 


PHASES. 


149 


FIG.  156. 


the  crests  at  c,  c,  c.     A  little  later,  when  either  crest  has  moved 

half  way  to  the  place  of  the  next  one, 

both  figures  will  have  become  reversed  ; 

the  centre  will  be  a  hillock,  the  troughs 

will  be  at  c,  c,  and  the  crests  at  the 

middle  points  between  them. 

Except  in  the  circular  arrangement  of  the  crests  and  troughs 
around  a  centre,  the  waves  of  the  foregoing  experiment  illustrate 
common  sea-waves.  They  constitute  a  system  of  elevations  and 
depressions  moving  along  the  surface  at  right  angles  to  the  line 
of  the  wave-crest. 

220.  Phases.— In  the  cross-section  (Fig.   156),  where  the 
waves  are  shown  in  profile,  any  particular  part  of  the  curve  is 
called  a  phase.     Different  phases  are  generally  unlike,  both  in  ele- 
vation and  in  movement.     The  corresponding  parts  of  different 
waves  are  called  like  phases;   and  those  points  in  which   the 
molecular  motions  are  reversed  are  called  opposite  jjhases.     The 
highest  points  of  the  crests  of  two  waves  are  like  phases ;  the 
highest  point  of  the  crest  and  the  lowest  point  of  the  trough  are 
opposite  phases.     Two  points  half  way  from  crest  to  trough,  one 
on  the  front  of  the  wave,  and  the  other  on  the  rear  of  it,  are  also 
opposite  phases,  although  they  are  at  the  same  elevation;  for  they 
are  moving  in  opposite  directions.     The  length  of  a  wave  is  the 
horizontal  distance  between  two  successive  like  phases. 

221.  Molecular  Movements. — The  water  which  constitutes 
a  system  of  waves  does  not  advance  along  the  surface,  as  the  waves 
themselves  do  ;  for  a  floating  body  is  not  borne  along  by  them,  but 
alternately  rises  and  falls  as  the  waves  pass  under  it.     Each  par- 
ticle of  water,  instead  of  advancing  with  the  wave,  oscillates  about 
its  mean  place,  alternately  rising  as  high  as  the  crest,  and  falling 
as  low  as  the  trough.     Its  path  is  the  circumference  of  a  vertical 
circle.     Let  B  B'  (Fig.  157)  represent  two  successive  troughs,  and 

FIG.  157. 


_      ,  _^_  ,  ____  =  _  __  ._^_  _.  _  _  _^._  . 


C  the  intervening  crest;  and  for  convenience  suppose  a  a',  the 
wave  length,  to  be  divided  into  eight  equal  parts.     The  waves 


150  HYDROSTATICS. 

move  in  the  direction  of  the  straight  arrow,  while  the  particles  of 
water  revolve  in  the  direction  of  the  bent  arrows.  The  points 
1,  2,  3,  &c.,  represent  particles  which,  if  the  water  were  at  rest, 
would  be  directly  above  the  points  a,  b,  c,  &c.  At  the  moment 
represented,  1  is  at  the  extreme  left  of  its  revolution,  2  is  at  45° 
below,  3  at  the  lowest  point,  &c.  When  the  wave  has  advanced 
one-eighth  of  its  length,  1  will  have  ascended  45°,  2  will  have  as- 
cended to  the  extreme  left,  and  each  of  the  eight  particles  will 
have  revolved  one-eighth  of  the  circumference  shown  in  the  figure. 
Then  4  will  be  at  the  bottom,  and  8  at  the  top.  Each  particle  of 
water  on  the  front  of  the  wave,  from  1  to  3,  and  from  7  to  3',  is 
ascending  ;  each  one  on  the  rear,  from  3  to  7,  is  descending.  It 
is  plain  that  while  the  wave  advances  its  whole  length,  that  is, 
while  the  phase  B  is  moving  to  B',  each  particle  makes  a  complete 
revolution;  3',  which  is  now  lowest,  will  be  lowest  again,  having 
in  the  meantime  occupied  all  other  points  of  the  circumference. 

Particles  below  the  surface,  as  far  as  the  wave  disturbance 
reaches,  perform  synchronous  revolutions,  but  in  smaller  circles, 
as  represented  in  the  figure. 

222.  Form  of  Waves  of  Oscillation.— The  sectional  form 
of  these  waves  is  that  of  the  inverted  trochoid,  a  curve  described 
by  a  point  in  a  circle  as  it  rolls  on  a  straight  line.     The  curvature 
of  the  crest  is  always  greater  than  that  of  the  trough,  and  the 
summit  may  possibly  be  a  sharp  ridge,  in  which  case  the  section 
of  the  trough  is  a  cycloid,  the  describing  point  of  the  rolling  circle 
being  on  the  circumference  ;  the  height  of  such  waves  is  to  their 
length  as  the  diameter  of  a  circle  to  the  circumference.     If  waves 
are  ever  higher  than  about  one-third  of  their  length,  the  summits 
are  broken  into  spray. 

223.  Distortion  of  the  Vertical  Columns.— Where  the 
surface  is  depressed  below  its  level,  some  of  the  water  must  be 
crowded  laterally  out  of  its  place,  and  the  vertical  columns,  being 
shorter,  must  necessarily  be  wider,  at  least  in  the  upper  part.    So, 
too,  where  the  surface  is  raised  above  its  level,  the  lengthened 
columns  must  be  narrower.     In  Fig.  157  these  effects  are  made 
apparent  as  the  necessary  result  of  the  revolutions  of  the  particles. 
The  dotted  lines,  1  a,  2  b,  3  c,  &c.,  were  all  vertical  lines  when  the 
water  was  at  rest.     But  now  they  are  swayed,  some  to  the  right 
and  some  to  the  left,  none  being  vertical,  except  under  the  highest 
and  lowest  points  of  the  waves.     Under  the  trough  the  lines  are 
spread  apart,  and  under  the  crest  they  are  drawn  together.    The 
sectional  figures  1  a  b  2,  2  b  c  3,  &c.,  which  would  all  be  rectangu- 
lar if  the  water  were  at  rest,  are  now  distorted  in  form,  the  upper 


SEA-WAVES.  151 

parts  being  alternately  expanded  and  contracted  in  breadth  as  the 
successive  phases  pass  them. 

224.  Sea- Waves. — The  waves  raised  by  the  wind  rarely  ex- 
hibit the  precise  forms  above  described,  and  the  particles  rarely 
revolve  in  exact  circles,  partly  because  there  is  scarcely  ever  a  sys- 
tem of  waves  undisturbed  by  other  systems,  which  are  passing 
over  the  water  at  the  same  time,  and  partly  because  the  wind, 
which  was  the  original  cause  of  the  waves,  acts  continually  upon 
their  surfaces  to  distort  and  confuse  them. 

The  interference  of  waves  denotes,  in  general,  the  resultant 
system,  which  is  produced  by  the  combination  of  two  or  more 
separate  systems.  The  joint  effect  of  two  systems  is  various,  ac- 
cording as  they  are  more  or  less  unlike  as  to  length  of  waves. 
But  even  if  two  systems  are  just  alike,  still  the  effect  of  interfer- 
ence will  vary,  according  to  the  coincidence  or  the  degree  of  dis- 
crepancy of  their  like  phases.  For  instance,  if  two  similar 
systems  exactly  coincide,  phase  for  phase,  the  waves  simply  have 
double  height ;  or,  in  general  terms,  there  is  double  intensity  in 
the  wave  motion.  But  if  the  phases  of  one  system  exactly  coincide 
with  the  opposite  phases  of  the  other,  then  the  water  is  nearly 
level,  the  crests  of  each  system  filling  the  troughs  of  the  other. 
These  two  effects  may  be  plainly  seen  in  the  intersections  of  ring- 
waves  formed  by  dropping  two  pebbles  on  still  water. 

225.  Waves  of  Translation. — The  principal  characteristics 
of  the  wave  of  translation  are,  that  it  is  solitary — i.  e.,  it  does  not 
belong  to  a  system,  like  the  other  kind ;  and  that  its  length  and 
velocity  both  depend  on  the  depth  of  the  water.     Where  the  water 
is  deeper,  the  wave  travels  faster,  and  its  length  (measured  in  the 
direction  of  its  progress)  is  longer.     A  wave  of  this  character  is 
started  in  a  canal  by  a  moving  boat ;  and  when  the  boat  stops,  it 
moves  on  alone.     A  grand  example  of  this  species  i»  found  in  the 
tide- wave  of  the  ocean.     It  is  called  the  wave  of  translation  be- 
cause the  particles  of  water  are  borne  forward  a  certain  distance 
while  the  wave  is  passing,  and  then  remain  at  rest. 


PART    III. 


CHAPTER    I. 

PROPERTIES    OF    GASES.— INSTRUMENTS    FOR   INVESTIGATION. 

226.  Gases  Distinguished  from   Liquids.— The  property 
of  mobility  of  particles,  which  belongs  to  all  fluids,  is  more  re- 
markable in  gases  than  in  liquids. 

While  gaseous  substances  are  compressed  with  ease,  they  are 
always  ready  to  expand  and  occupy  more  space.  This  property, 
called  dilat ability,  scarcely  belongs  to  liquids  at  all. 

This  property  may  be  experimentally  illustrated  by  placing  a 
bag  only  partly  full  of  air  under  the  receiver  of  an  air  pump  and 
exhausting  the  air ;  the  external  pressure  having  been  removed, 
the  bag  will  seem  full  almost  to  bursting,  the  contained  air  hav- 
ing dilated  to  many  times  its  former  volume. 

Invert  a  flask  containing  air  into  a  beaker  of  colored  water, 
and  place  the  whole  under  the  receiver  of  an  air  pump.  As  the 
air  is  exhausted  the  contained  air  in  the  flask  will  expand  and, 
driving  the  water  out  of  the  neck  of  the  flask,  will  rise  in  bubbles 
to  the  surfaqp.  Upon  admitting  air  again  to  the  receiver,  the 
water  will  be  forced  into  the  flask  to  take  the  place  of  the  escaped 
air,  and  will  rise  until  the  tension  of  the  contained  air,  together 
with  the  weight  of  the  water  column,  is  equal  to  that  of  the  air  in 
the  receiver. 

227.  Tension  of  Gases. — By  the  term  tension  just  used,  is 
meant  the  force  exerted  by  the  gas  at  each  instant  in  opposition 
to  any  compressing  or  restraining  force  ;  or,  in  other  words,  the 
force  of  expansion.    The  molecules  of  the  gas  are  supposed  to  be 
flying  through  space  with  great  velocity  in  straight  lines.     The 
combined  effect  of  the  impact  of  these  molecules  upon  the  walls 
of  the  containing  vessel  is  an  outward  pressure  which  is  opposed 
bv  the  strength  of  the  material  of  the  vessel.     Tn  the  first  experi- 


OSMOSE    OF    GASES. 


153 


merit  given,  the  impact  of  the  molecules  of  the  air  in  the  room 
upon  the  outside  of  the  bag  counterbalanced  the  impact  of  the 
molecules  of  the  contained  air  upon  the  inside;  but  when  the 
external  air  was  removed,  there  was  no  counterbalancing  force, 
until  the  bag  expanded  so  much  that  the  strength  of  elasticity  of 
the  rubber  itself  equaled  the  resultant  of  the  impacts  within. 
This  theory  of  tension  will  be  of  great  help  in  discussing  the  sub- 
ject of  expansion  by  heat. 

228.  Change  of  Condition.— Liquids,  and  even  solids,  may 
be  changed  into  the  gaseous  or  aeriform  condition  by  heating 
them  sufficiently.     By  being  cooled,  they  return  again  to  their 
former  state.     In  the  gaseous  form  they  are  called  vapors.    All 
substances  which  are  ordinarily  gases  can  be  so  far  cooled,  espe- 
cially under  great  pressure,  as  to  be  reduced  to  the  liquid  or  solid 
form. 

Those  which  can  only  be  thus  reduced  under  very  great  pres- 
sures, and  at  very  low  temperatures,  are  regarded  as  types  of  a 
theoretically  perfect  gas. 

229.  Diffusion  of  Gases.— If  two  flasks,  A  and  B,  be  con- 
nected by  a  tube,    as  in  Fig.   158,  and   the  pm   159 
upper,  At  be  filled  with  hydrogen,  or  illumi- 
nating gas,  and  the  lower  B  with  carbonic  diox-  pIG 

ide,  after  a  time  some  of  the  lighter  gas  will  be 
found  in  J3,  having  passed  down  through  the 
tube,  while  a  part  of  the  heavy  gas  in  B  will 
have  passed  upwards  to  A.  This  result  must 
follow  from  the  theory  of  molecular  motion 
given  before.  The  action  is  called  diffusion. 

230.  Osmose  of  Gases. — Cement  a  glass 
tube,  about  twenty-four  inches  long,  to  a  por- 
ous cell  B  (Fig.  159),  and  dip  the  lower  end 
of  the  tube  into  colored  liquid  A.    Now  fill  an 
inverted  bell  jar  with  hydrogen,  or  illuminat- 
ing gas,  and  place  it  over  B.     The  gas  will 
make  its  way  into  the  porous  cell  more  rapidly 
than  the  air  makes  its  way  out,  diffusion  in- 
wards being  more  rapid  than   diffusion  out- 
wards, and,  in  consequence,  some  air  will  be 
driven  out  through  the  glass  tube,  escaping  in 
bubbles  through  the  liquid  in  A.     Upon  re- 
moving the  bell  jar  the  gas  within  the  porous 
cell  will  pass  out  again  more  rapidly  than  air 
can  pass  in,  and  a  partial  vacuum  will  be 


154  PNEUMATICS. 

i'ormed,  causing  a  rise  of  the  colored  liquid  in  the  tube.  This 
mixing  of  gases  through  a  porous  cell,  or  a  thin  moistened  mem- 
brane, is  denominated  Osmose  of  Gases  to  distinguish  it  from  a 
similar  action  of  liquids  under  like  conditions.  The  inward  flow 
is  termed  endosmose,  and  the  outward  flow  exosmose,  distinctions 
which  are  of  no  great  importance. 

231.  Weight  of  Gases. — Like  all  other  forms  of  matter, 
gases  have  weight.  Some  are  relatively  light,  some  heavy.  Take 
a  copper  globe,  and  hang  it  upon  one  scale'  pan  of  a  delicate 
balance,  and  accurately  counterpoise  it.  Next  exhaust  the  air 
from  the  globe,  and  it  will  be  found  lighter  than  before ;  fill  with 
carbonic  dioxide,  and  it  will  weigh  much  more  than  at  first.  The 
heavier  gases  may  be  poured  from  one  vessel  to  another  like 
water ;  carbonic  dioxide  may  be  poured  from  a  beaker  upon  a 
burning  candle,  which  may  thus  be  extinguished. 

232. — Pressure  of  Gases.— As  a  consequence  of  the  weight 
of  gases  we  have  to  consider  the  pressure  exerted  by  them.  We 
shall  use  the  atmosphere  as  a  type  of  all  gases.  Across  the  open 
top  of  a  cylindric  receiver  stretch  a  sheet  of  rubber ;  upon  ex- 
hausting the  air  from  the  receiver,  the  rubber  will  be  pressed  in- 
wards by  the  external  air.  Substitute  for  the  sheet  of  rubber  a 
sheet  of  wetted  bladder,  which  allow  to  dry.  Upon  exhausting 
the  air  the  bladder  will  burst,  under  the  pressure  inwards,  with  a 
loud  report. 

Exhaust  the  air  from  a  receiver,  into  which  projects  a  jet  tube 
closed  with  a  stop-cock  ;  upon  submerging  the  outer  end  of  the 
jet  and  opening  the  stop-cock,  a  fountain  in  vacuo  will  be  pro- 
duced. 

Exhaust  the  air  from  two  closely  fitted  hemispheres,  called 
Magdeburg  hemispheres,  of  about  four  inches  diameter  ;  a  force 
of  over  175  Ibs.  will  be  required  to  separate  them. 

Having  a  cylinder  about  five  inches  in  diameter,  with  closely 
fitted  piston,  attach  a  weight  of  250  Ibs.  to  the  lower  side  of  the 
piston,  exhaust  the  air  from  the  cylinder  above  the  piston,  and 
the  weight  will  be  raised. 

The  pressure  of  the  air  upon  our  bodies  and  the  outward  pres- 
sure of  the  blood  against  the  walls  of  the  small  veins  and  capil- 
laries are  in  equilibrium.  Place  the  palm  of  the  hand  upon  the 
broad  opening  of  a  receiver,  called  a  hand  glass,  and  exhaust  the 
air  beneath ;  the  air  pressure  being  removed,  the  flesh  will  pro- 
trude into  the  receiver,  and  the  skin,  by  its  redness,  will  give 
evidence  of  the  engorgement  of  the  blood-vessels. 

233.  Buoyancy. — When  the  water  displaced  by  an  immersed 


TORRICELLI'S    EXPERIMENT. 


155 


Oody  weighs  more  than  the  body  itself,  it  will  rise  to  the  surface 
and  float ;  so  too  when  the  volume  of  air  displaced  by  a  body 
weighs  more  than  that  body,  it  will  rise  and  float  in  the  air.  At- 
tach the  gas  jet  to  a  clay  pipe  by  rubber  tubing,  and  blow  soap- 
bubbles  ;  these  will  rise  rapidly  to  the  ceiling  of  the  room.  Blow 
a  small  bubble,  and  then  transfer  the  end  of  the  tube  to  the  gas 
jet  and  enlarge  the  bubble  till  its  specific  gravity  is  about  the 
same  as  that  of  the  air ;  it  will  now  float  about  the  room,  some- 
times rising,  sometimes  falling,  until  it  bursts. 

Large  balloons  have  ascended  to  a  height  of  seven  miles. 

If  a  large  and  a  small  body  are  in  equilibrium  on  the  two  arms 
of  a  balance,  and  the  whole  be  set  under  a  receiver,  and  the  air 
be  removed,  the  larger  body  will  preponderate,  showing  that  it  is 
really  the  heaviest.  Their  apparent  equality  of  weight  when  in 
the  air  is  owing  to  its  buoyant  power,  which  diminishes  the 
apparent  weight  of  an  immersed  body  by  just  the  weight  of  the 
displaced  fluid.  Hence,  the  larger  the  body,  the  more  weight  it 
loses. 

If  the  air  be  exhausted  from  a  tube,  four  or  five  feet  long  and 
two  inches  in  diameter,  containing  a  small  coin  and  a  feather,  it 
will  be  found,  upon  quickly  inverting  the  tube,  that  the  coin  and 
the  feather  will  fall  through  its  length  in  the  same  time,  both 
having  the  same  velocity  ;  if  it  were  not 
for  the  obstruction  of  the  air,  all  bodies  FIG.  160. 

would  fall  to  the  earth  with  the  same 
velocity. 

234.  Torricelli's  Experiment.— 
A  glass  tube  A  B  (Fig.  160)  about  three 
ieet  long,  and  hermetically  sealed  at 
one  end,  is  filled  with  mercury,  and 
then,  while  the  finger  is  held  tightly  on 
the  open  end,  it  is  inverted  in  a  cup  of 
mercury.  On  removing  the  finger  after 
the  end  of  the  tube  is  beneath  the  sur- 
face of  the  mercury,  the  column  sinks 
a  little  way  from  the  top,  and  there  re- 
mains. Its  height  is  found  to  be  nearly 
thirty  inches  above  the  level  of  mercury 
in  the  cup.  If  sufficient  care  is  taken  to 
expel  globules  of  air  from  the  liquid, 
the  space  above  the  column  in  the 
tube  is  as  perfect  a  vacuum  as  can  be 
obtained.  It  is  called  the  Torricellian 
vacuum,  from  Torricelli  of  Italy,  a 


156  PNEUMATICS. 

disciple  of  Galileo,  who,  by  this  experiment,  disproved  the  doctrine 
that  nature  abhors  a  vacuum,  and  fixed  the  limits  of  atmospheric 
pressure. 

235.  Pressure  of  Air  Measured. — The  column  is  sus- 
tained in  the  Torricellian  tube  by  the  pressure  of  air  on  the  sur- 
face of  mercury  in  the  vessel  ;  for  the  level  of  a  fluid  surface 
cannot  be  preserved  unless  there  is  an  equal  pressure  on  every 
part.    Hence,  the  column  of  mercury  on  one  part,  and  the  column 
of  air  on  every  other  equal  part,  must  press  equally.    To  determine, 
therefore,  the  pressure  of  air,  we  have  only  to  weigh  the  column 
of  mercury,  and  measure  the  area  of  the  mouth  of  the  tube.     If 
this  is  carefully  done,  it  is  found  that  the  weight  of  mercury  is  about 
14.7  Ibs.  on  a  square  inch.     Therefore  the  atmosphere  presses  on 
the  earth  with  a  force  of  nearly  15  pounds  to  every  square  inch, 
or  more  than  2000  Ibs.  per  square  foot. 

The  specific  gravity  of  mercury  is  about  13. 6;  and  therefore 
the  height  of  a  column  of  water  in  a  Torricellian  tube  should  be 
13.6  times  greater  than  that  of  mercury,  that  is,  about  34  feet. 
Experiment  shows  this  to  be  true.  And  it  was  this  significant 
fact,  that  equal  iveights  of  water  and  mercury  are  sustained  in 
these  circumstances,  which  led  Torricelli  to  attribute  the  effect  to 
a  common  force,  namely,  the  pressure  of  the  air. 

236.  Pascal's  Experiment. — As  soon  as  Torricelli's  discov- 
ery was  known,  Pascal  of  France  proposed  to  test  the  correct- 
ness of  his  conclusion,  by  carrying  the  apparatus  to  the  top  of  a 
mountain,  in  order  to  see  if  less  air  above  the  instrument  sustained 
the  mercury  at  a  less  height.     This  was  found  to  be  true;  the 
column  gradually  fell,  as  greater  heights  were  attained.     The  ex- 
periment of  Pascal  also  determined  the  relative  density  of  mer- 
cury and  air.     For  the  mercury  falls  one-tenth  of  an  inch  in 
ascending  87.2  feet ;  therefore  the  weight  of  the  one-tenth  of  an 
inch  of  mercury  was  balanced  by  the  weight  of  the  87.2  feet  of 
air.    Therefore  the  specific  gravities  of  mercury  and  air  (being  in- 
versely as  the  heights  of  columns  in  equilibrium)  are  as  (87.2  x  12 
x  10  =)  10464  :  1.     In  the  same  way  it  is  ascertained  that  water 

is  770  times  as  dense  as  air.  These  results  can  of  course  be  con- 
firmed by  directly  weighing  the  several  fluids,  which  could  not  be 
done  before  the  invention  of  the  air-pump. 

That  it  is  the  atmospheric  pressure  which  sustains  the  column 
of  mercury  may  be  shown  thus :  Place  the  Torricellian  tube  and 
cistern  under  a  receiver,  made  for  the  purpose,  and  exhaust  the 
air ;  the  mercury  will  fall  lower  and  lower  at  each  stroke  of  the 
pump,  until,  if  the  pump  be  in  good  working  order,  the  column 
will  be  nearly  at  the  level  of  the  mercury  in  the  cistern. 


MARIOTTE'S    LAW. 


157 


FIG.  161. 


FIG.  162. 


237.  Mariotte's  Law.— 

At  a  given  temperature,  the  volume  of  air  is  inversely  as  the 
compressing  force. 

An  instrument  constructed  for  showing  this  is  called  Mariotte's 
tube.  The  end  B  (Fig.  161)  is  sealed,  and  A  open.  Pour  in  small 
quantities  of  mercury,  inclining  the  tube  so  as  to  let  air  in  or  out, 
till  both  branches  are  filled  to  the  zero  point.  The  air  in  the 
short  branch  now  has  the  same  tension  as  the  external  air,  since 
they  just  balance  each  other.  If  mercury  be  poured  in  till  the 
column  in  the  short  tube  rises  to  (7,  the  inclosed  air  is  reduced  to 
one-half  of  its  original  volume,  and  the  column  A  in  the  long  branch 
is  found  to  be  29  or  30  inches  above  the  level  of  C,  according  to 
the  barometer  at  the  time.  Thus,  two  atmospheres,  one  of  mer- 
cury, the  other  of  air  above  it,  have  compressed  the  inclosed  air 
into  one-half  its  volume.  If  the 
tube  is  of  sufficient  length,  let 
mercury  be  poured  in  again,  till 
the  air  is  compressed  to  one- 
third  of  its  original  space;  the 
long  column,  measured  from 
.the  level  of  the  mercury  in  the 
short  one,  is  now  twice  as  high 
as  before;  that  is,  three  atmos- 
pheres, two  of  mercury  and  one 
of  air,  have  reduced  the  same 
quantity  of  air  to  one-third  of  its 
first  volume.  This  law  has  been 
found  to  hold  good  in  regard  to 
atmospheric  air  up  to  a  pressure 
of  near]y  thirty  atmospheres. 

On  the  other  hand,  if  the 
pressure  on  a  given  mass  of  air 
is  diminished,  its  volume  is 
found  to  increase  according  to 
the  same  law.  When  the  pres- 
sure is  half  an  atmosphere,  the 
volume  is  doubled ;  when  one- 
third  of  an  atmosphere,  the 
volume  is  three  times  as  great, 
&c. 

This  may  be  shown  by  filling 
a  tube  A,  closed  at  one  end,  nearly  full  of  mercury,  and  inverting 
it  in  a  tube-like  cistern  B,  as  in  Fig.  162.  Suppose  that  the  tube, 
of  uniform  bore,  contains  an  inch  of  air  when  the  mercury  is  at 


158  PNEUMATICS. 

the  same  level  within  and  without ;  upon  raising  the  tube  until 
the  column  within  is  about  fifteen  inches  high  the  air  will  be 
found  to  occupy  two  inches  of  space.  At  the  beginning  of  the 
experiment  the  mercury  being  at  the  same  level  in  both  tubes,  the 
pressure  upon  the  contained  air  was  due  to  the  atmosphere,  and 
was  about  14.7  Ibs.  per  square  inch.  At  the  close  the  pressure 
was  less  by  the  force  necessary  to  sustain  the  fifteen  inches  of 
mercury,  leaving  only  a  pressure  of  about  one-half  an  atmosphere. 
Experiments  in  this  case  will  not  be  satisfactory  unless  precau- 
tions be  taken  to  remove  the  air  bubbles  which  adhere  to  the  glass 
tube.  Since  the  tension  of  the  inclosed  air  always  balances  the 
compressing  force,  and  since  the  density  is  inversely  as  the 
volume,  it  follows  from  Mario tte's  Law  that,  when  the  tempera- 
ture is  the  same, 

The  tension  of  air  varies  as  the  compressing  force ;  and  TJie 
tension  of  air  varies  as  its  density. 

This  law  is,  however,  not  strictly  true  of  gases,  except  when 
far  removed  from  the  critical  point ;  when  near  the  point  of  con- 
densation the  departure  from  the  law  is  most  marked.  Even  in 
the  ordinary  state  the  law  is  not  strictly  followed  by  many  gases. 

238.  Dalton's  Law.— 

At  a  given  temperature  the  tension  of  a  mixture  of  gases  is 
equal  to  the  sum  of  the  tensions  of  the  gases  taken  separately. 

In  this  law  a  mixture  is  spoken  of  and  not  a  chemical  combi- 
nation. Each  gas  diffuses  and  is  found  equally  distributed 
throughout  the  containing  vessel  just  as  though  no  other  gas  were 
present,  differing  in  this  respect  from  mechanical  mixtures  of 
liquids,  such  as  oil  and  water,  in  which  the  components  of  the 
mixture  arrange  themselves  according  to  their  specific  gravities. 
If  a  vessel  A  contains  a  cubic  foot  of  nitrogen  at  a  tension  of  ten 
Ibs.  per  square  inch,  and  a  perfect  vacuum  B  of  the  same  capacity 
be  connected  with  this,  and  all  the  gas  be  transferred  to  the  new 
vessel  By  the  tension  in  the  latter  case  will  be  the  same  as  in  the 
former,  ten  Ibs.  Now,  if  a  cubic  foot  of  oxygen  at  a  tension  of 
ten  Ibs.  be  transferred  to  the  vessel  B  also,  it  will  exert  a  pressure 
of  ten  Ibs.  just  as  though  the  nitrogen  were  not  present,  giving  a 
total  pressure  of  twenty  Ibs.  for  the  mixture.  These  two  illustra- 
tions have  been  given  to  prevent  any  misapprehension  which  may 
arise  from  the  following  frequently  repeated  and  very  concise 
wording  of  the  law  :  "  Every  gas  acts  as  a  vacuum  with  respect  to 
every  other." 

239.  Laws  of  Mixture  of  Gases  and  Liquids.— Water 

and  many  other  liquids  will  contain  gases  in  solution  ;  but  under 
the  same  conditions  of  temperature  and  pressure,  a  given  liquid 


MIXTURE    OF    GASES    AND    LIQUIDS.  159 

does  not  absorb  equal  quantities  of  different  gases.  For  example, 
at  the  mean  temperature  and  pressure  water  dissolves  about  .025 
of  its  volume  of  nitrogen,  .046  of  its  volume  of  oxygen,  its  own 
volume  of  carbonic  dioxide,  and  430  times  its  volume  of  am- 
monia. Mercury,  on  the  other  hand,  does  not  dissolve  any  of 
the  gases. 

Experiment  has  determined  the  three  following  laws  of  the 
mixture  of  liquids  and  gases. 

1st.  For  the  same  gas,  the  same  liquid,  at  a  constant  tempera- 
ture,  the  weight  of  gas  absorbed  is  proportional  to  the  pressure. 
From  this  it  follows  that  the  volume  dissolved  is  constant,  what- 
ever may  be  the  changes  in  pressure,  or,  what  is  the  same  thing, 
the  density  of  the  gas  absorbed  bears  a  constant  ratio  to  that  of 
the  gas  not  absorbed. 

2d.  The  quantity  of  gas  absorbed  increases  as  the  temperature 
decreases. 

3d.  The  quantity  of  a  gas  which  a  given  liquid  will  dissolve  is 
independent  of  the  kinds  and  quantities  of  other  gases  which  may 
already  be  held  in  solution. 

If  instead  of  a  single  gas  in  contact  with  the  liquid,  a  mix- 
ture of  several  gases  be  used,  each  of  these  will  be  dissolved  in  the 
quantity  due  to  its  proportional  part  of  the  total  pressure.  For 
example,  since  oxygen  forms  only  about  one-fifth  the  volume  of 
the  air,  water  under  ordinary  conditions  absorbs  the  same  quantity 
of  oxygen,  as  if  the  atmosphere  were  wholly  of  oxygen  under 

a  pressure  of  — ~  Ibs. 
o 

The  first  law  may  be  experimentally  illustrated  by  opening  a 
bottle  of  common  soda  water.  As  soon  as  the  cork  is  loosened 
it  is  driven  out  by  the  tension  of  the  confined  carbonic  dioxide 
above  the  liquid,  and  the  pressure  being  reduced  by  the  escape  of 
the  free  gas,  the  absorbed  gas  is  at  once  given  off  in  bubbles, 
the  escape  of  which  produces  foaming. 

If  after  all  the  gas  has  seemingly  escaped,  a  portion  of  the 
liquid  be  poured  into  a  beaker  and  placed  under  the  receiver  of 
the  air  pump,  a  fresh  discharge  of  bubbles  will  follow  the  first 
stroke  of  the  pump,  consequent  upon  the  still  further  reduction 
of  pressure. 

A  portion  poured  into  a  flask  and  heated  will  serve  to  illustrate 
the  second  law,  the  rise  in  temperature  causing  a  constant  rise  of 
bubbles  of  gas. 

240.  The  Barometer. — When  the  Torricellian  tube  and 
basin  are  mounted  in  a  case,  and  furnished  with  a  graduated  scale, 
the  instrument  is  called  a  barometer.  The  scale  is  divided  into 


160  PNEUMATICS. 

inches  and  tenths,  and  usually  extends  from  26  to  32  inches,  a 
space  more  than  sufficient  to  include  all  the  natural  variations  in 
the  weight  of  the  atmosphere.  By  attaching  a  vernier  to  the 
scale,  the  reading  may  be  carried  to  hundredths  and  thousandths 
of  an  inch,  as  is  commonly  done  in  meteorological  observations. 
By  observing  the  barometer  from  day  to  day,  and  from  hour  to 
hour,  it  is  found  that  the  atmospheric  pressure  is  constantly 
fluctuating. 

As  the  meteorological  changes  of  the  barometer  are  all  com- 
prehended within  a  range  of  two  or  three  inches,  much  labor  has 
been  expended  in  devising  methods  for  magnifying  the  motions 
of  the  mercurial  column,  so  that  more  delicate  changes  of  atmos- 
pheric pressure  might  be  noted.  The  inclined  tube  and  the  wheel 
barometer  are  intended  for  this  purpose.  A  description  of  these 
contrivances,  however,  is  unnecessary,  as  they  are  all  found  to  be 
inferior  in  accuracy  to  the  simple  tube  and  basin. 

241.  Corrections  for  the  Barometer. — 

1.  For  change  of  level  in  the  basin. — The  numbers  on  the 
barometer  scale  are  measured  from  a  certain  zero  point,  which  is 
assumed  to  be  the  level  of  the  mercury  in  the  basin.     If  now  the 
column  falls,  it  raises  the  surface  in  the  basin ;  and  if  it  rises,  it 
lowers  it.    If  the  basin  is  broad,  the  change  of  level  is  small,  but 
it  always  requires  a  correction.     To  avoid  this  source  of  error,  the 
bottom  of  the  basin  is  made  of  flexible  leather,  with  a  screw  under- 
neath it,  by  which  the  mercury  may  be  raised  or  lowered,  till  its 
surface  touches  an  index  that  marks  the  zero  point.    This  adjust- 
ment should  always  be  made  before  reading  the  barometer. 

2.  For  capillarity. — In  a  glass  tube  mercury  is  depressed  by 
capillary  action  (Art.   200).    The  amount  of  depression  is  less  as 
the  tube  is  larger.     This  error  is  to  be  corrected  by  the  manufac- 
turer, the  scale  being  put  below  the  true  height  by  a  quantity  equal 
to  the  depression. 

There  is  a  slight  variation  in  this  capillary  error,  arising  from 
the  fact  that  the  rounded  summit  of  the  column,  called  the  menis- 
cus, is  more  convex  when  ascending  than  when  descending.  To 
render  the  meniscus  constant  in  its  form,  the  barometer  should  be 
jarred  before  each  reading. 

3.  For  temperature. — As  mercury  is  expanded  by  heat  and 
contracted  by  cold,  a  given  atmospheric  pressure  will  raise  the 
column  too  high,  or  not  high  enough,  according  to  the  tempera- 
ture of  the  mercury.    A  thermometer  is  therefore  attached  to  the 
barometer,  to  show  the  temperature  of  the  instrument.    By  a  table 
of  corrections,  each  reading  is  reduced  to  the  height  the  mercury 
would  have  if  its  temperature  was  32°  F. 


THE    ANEROID    BAROMETER. 


161 


FIG.  163. 


FIG.  164. 


s| 

si 


4.  For  altitude  of  station. — Before  comparing  the  observations- 
of  different  places,  a  correction  must  be  made  for  altitude  of  sta- 
tion, because  the  column  is  shorter  according  as  the  place  is  higher 
above  the  sea  level. 

242.  The  Aneroid  Barometer.— This  is  a  small  and  port- 
able instrument,  in  appearance  a  little  like  a  large  chronometer. 
The  essential  part  .of  this  barometer  is  a  flat  cylindrical  metallic 
box,  shown  in  section  at  A   (Fig.   163), 

whose  upper  surface  is  corrugated,  so  as 
to  be  yielding. 

The  box  being  partly  exhausted  of  air, 
the  external  pressure  causes  the  top  to 
sink  in  to  a  certain  extent ;  if  the  pres- 
sure increases,  the  surface  descends  a 
little  more  ;  if  it  diminishes,  a  little  less. 

These  small  movements  are  communicated  to  a  lever  I  (Fig. 
163  )  whose  end  c  moves  over  a  scale  of  .inches  on  the  case  (Fig. 
164).  To  insure  contact  of  the  pin  o 
(Fig.  163),  and  also  to  secure  a  constant 
resistance  to  the  motion  of  the  lever  b  a 
spring  d  is  attached  to  the  lever  b  near 
the  fulcrum,  which  is  pressed  upon  by  the 
screw  s,  whose  graduated  head  H  is  of 
the  same  diameter  as  the  box.  The  end 
of  this  spring  e  (Fig.  164)  must  be 
brought  to  coincide  with  the  end  of  the 
lever,  by  turning  the  screw-head  H.  The 
reading  of  inches  and  tenths  is  taken 

from  the  scale  of  inches,  the  hundredths  and  thousandths  being 
given  by  the  screw-head.  This  is  one  of  the  most  simple  of  all 
the  aneroids  in  construction.  A  table  of  corrections  for  tempera- 
ture, and  reductions  to  the  standard  mercurial  barometer  is  entered 
upon  the  cover. 

243.  Pressure  and  Latitude. — The  mean  pressure  of  the 
atmosphere  at  the  level  of  the  sea  is  very  nearly  30  inches.     But 
it  is  not  the  same  at  all  latitudes.    From  the  equator  either  north- 
ward or  southward,  the  mean  pressure  increases  to  about  latitude 
30°,  by  a  small  fraction  of  an  inch,  and  thence  decreases  to  about 
65°,  where  the  pressure  is  less  than  at  the  equator,  and  beyond 
that  it  slightly  increases.     This  distribution  of  pressures  in  zones 
;is  due  to  the  great  atmospheric  currents,  caused  by  heat  in  con- 
nection with  the  earth's  rotation  on  its  axis. 

The  amount  of  variation  in  barometric  pressure  is  very  unequal 
in  different  latitudes  ;  and  in  general,  the  higher  the  latitude,  the 
11 


162  PNEUMATICS. 

greater  the  variation.  Within  the  tropics  the  extreme  range 
scarcely  ever  exceeds  one-fourth  of  an  inch,  while  at  latitude  40° 
it  is  more  than  two  inches,  and  in  higher  latitudes  even  reaches 
three  inches. 

244.  Diurnal  Variation. — If  a  long  series  of  barometric  ob- 
servations be  made,  and  the  mean  obtained  for  each  hour  of  the 
day,  the  changes  caused  by  weather  become  eliminated,  and  the 
diurnal  oscillation  reveals  itself.  It  is  found  that  the  pressure 
reaches  a  maximum  and  a  minimum  twice  in  24  hours.  The 
times  of  greatest  pressure  are  from  9  to  11,  and  of  least  pressure 
from  3  to  5,  both  A.  M.  and  P.  M.  In  tropical  climates  this  varia- 
tion is  very  regular,  though  small ;  but  in  the  temperate  zones  the 
irregular  fluctuations  of  weather  conceal  it  in  a  great  degree. 

This  daily  fluctuation  of  the  barometer  is  caused  by  the 
changes  which  take  place  from  hour  to  hour  of  the  day  in  the 
temperature,  and  by  the  varying  quantity  of  vapor  in  the  atmos- 
phere. 

The  surface  of  the  globe  is  always  divided  into  a  day  and  night 
hemisphere,  separated  by  a  great  circle  which  revolves  with  the 
sun  from  east  to  west  in  twenty-four  hours.  The  hemisphere  ex- 
posed to  the  sun  is  warm,  the  other  is  cold.  The  time  of  greatest 
heat  is  not  at  noon,  when  the  sun  is  in  the  meridian,  but  about 
two  or  three  hours  after  ;  the  period  of  greatest  cold  occurs  about 
four  in  the  morning.  As  the  hemisphere  under  the  sun's  rays 
becomes  heated,  the  air,  expanding  upwards  and  outwards,  flows 
over  upon  the  other  hemisphere  where  the  air  is  colder  and  denser. 
There  thus  revolves  round  the  globe  from  day  to  day  a  wave 
of  heated  air,  from  the  crest  of  which  air  constantly  tends  to  flow 
towards  the  meridian  of  greatest  cold  on  the  opposite  side  of  the 
the  globe. 

The  barometer  is  influenced  to  a  large  extent  by  the  elastic 
force  of  the  vapor  of  water  invisibly  suspended  in  the  atmosphere, 
in  the  same  way  as  it  is  influenced  by  the  dry  air  (oxygen  and 
hydrogen).  But  the  vapor  of  water  also  exerts  a  pressure  on  the 
barometer  in  another  way.  Vapor  tends  to  diffuse  itself  equally 
through  the  air  ;  but  as  the  particles  of  air  offer  an  obstruction  to 
the  watery  particles,  about  9  or  10  A.  M.,  when  evaporation  is 
most  rapid,  the  vapor  is  accumulated  or  pent  up  in  the  lower 
stratum  of  the  atmosphere,  and  being  impeded  in  its  ascent  its 
elastic  force  is  increased  by  the  reaction,  and  the  barometer  con- 
sequently rises.  When  the  air  falls  below  the  temperature  of  the 
dew-point  part  of  its  moisture  is  deposited  in  dew,  and  since  some 
time  must  elapse  before  the  vapor  of  the  upper  strata  can  diffuse 
itself  downwards  to  supply  the  deficiency,  the  barometer  falls, — 


HEIGHTS    MEASURED    BY    THE    BAROMETER.    163 

most   markedly   at   10   P.    M.,    when   the   deposition   of   dew   is 
greatest. 

245.  The  Barometer  and  the  Weather.— The  changes 
in  the  height  of  the  barometer  column  depend  directly  on  nothing 
else  than  the  atmospheric  pressure.   But  these  changes  of  pressure 
are  due  to  several  causes,  such  as  wind  and  changes  of  temperature 
and  moisture. 

The  practice  formerly  prevailed  of  engraving  at  different  points 
of  the  barometer  scale  several  words  expressive  of  states  of  weather, 
"  fair,  rain,  frost,  wind,"  &c.  But  such  indications  arc  worthless, 
being  as  often  false  as  true  ;  this  is  evident  from  the  fact  that  the 
height  of  the  column  would  be  changed  from  one  kind  of  weather 
to  another  by  simply  carrying  the  instrument  to  a  higher  or  lower 
station. 

No  general  system  of  rules  can  be  given  for  anticipating  changes 
of  weather  by  the  barometer,  which  would  be  applicable  in  differ- 
ent countries.  Eules  found  in  English  books  are  of  very  little 
value  in  America. 

Severe  and  extensive  storms  are  almost  always  accompanied 
by  a  fall  of  the  barometer  while  passing,  and  succeeded  by  a  rise 
of  the  barometer. 

246.  Heights  Measured  by  the  Barometer. — Since  mer- 
cury is  10464  times  as  heavy  as  air  (Art.  236),  if  the  barometer  is 
carried  up  until  the  mercury  falls  one  inch,  it  might  be  inferred 
that  the  ascent  is  10464  inches,  or  872  feet.     This  would  be  the 
case  if  the  density  were  the  same  at  all  altitudes.    But,  on  account 
of  diminished  pressure,  the  air  is  more  and  more  expanded  at 
greater  heights.    Besides  this,  the  height  due  to  a  given  fall  of  the 
mercury  varies  for  many  reasons,  such  as  the  temperature  of  the 
air,  the  temperature  of  the  mercury,  the  elevation  of  the  stations, 
and  their  latitude.     Hence,  the  measurement  of  heights  by  the 
barometer  is  somewhat  troublesome,  and  not  always  to  be  relied 
Ton.      Formulae  and  tables  for  this  purpose  are  to  be  found  in 
practical  works  on  physics. 

247.  The  Gauge  of  the  Air- Pump.— The  Torricellian  tube 
is  employed  in  different  ways  as  a  gauge  for  the  air-pump,  to  indi- 
cate the  degree  of  exhaustion.     In  Fig.  166  the  gauge  G  is  a  tube 
about  33  inches  long,  both  ends  of  which  are  open,  the  lower  im- 
mersed in  a  cup  of  mercury,  and  the  upper  communicating  with 
the  interior  of  the  receiver.     As  the  exhaustion  proceeds,  the 
pressure  is  diminished  within  the  tube,  and  the  external  air  raises 
the  mercury  in  it.     A  perfect  vacuum  would  be  indicated  by  a 
height  of  mercury  equal  to  that  of  the  barometer  at  the  time. 


164 


PNEUMATICS. 


Another  kind  of  gauge  is  a  barometer  already  filled,  the  basin 
of  which  is  open  to  the  receiver.  As  the  tension  of  air  in  the 
receiver  is  diminished,  the  column  descends,  and  would  stand  at  the 
same  level  in  both  tube  and  basin,  if  the  vacuum  were  perfect. 

A  modified  form  of  the  last,  called  the  siphon  gauge, 
is  the  best  for  measuring  the  rarity  of  the  air  in  the 
receiver  when  the  vacuum  is  nearly  perfect.  Its  con- 
struction is  shown  by  Fig.  165.  The  top  of  the  column, 
A,  is  only  5  or  6  inches  above  the  level  of  B  in  the  other 
branch  of  the  recurved  tube.  As  the  air  is  withdrawn 
from  the  open  end  (7,  the  tension  at  length  becomes  too 
feeble  to  sustain  the  column  ;  it  then  begins  to  descend, 
and  the  mercury  in  the  two  branches  approaches  a  com- 
mon level. 


CHAPTER     II. 

INSTRUMENTS  WHOSE  OPERATION  DEPENDS  ON  THE  PROPER. 

TIES    OF    AIR. 

248.  The  Air-Pump. — This  is  an  instrument  by  which 
nearly  all  the  air  can  be  removed  from  a  vessel  or  receiver.  It  has 
a  variety  of  forms,  one  of  which  is  shown  in  Fig.  166.  In  the 

FIG.  166. 


THE    AIR-PUMP. 


165 


FIG.  167. 


barrel  B  an  air-tight  piston  is  alternately  raised  and  depressed  by 
the  lever,  the  piston-rod  being  kept  vertical  by  means  of  a  guide. 
The  pipe  P  connects  the  bottom  of  the  barrel  with  the  brass  plate 
L,  on  which  rests  the  receiver  R.  The  surface  of  the  plate  and 
the  edge  of  the  receiver  are  both  ground  to  a  plane.  G  is  the 
gauge  which  indicates  the  degree  of  exhaustion.  There  are  three 
valves,  the  first  at  the  bottom  of  the  barrel,  the  second  in  the 
piston,  and  the  third  at  the  top  of  the  barrel.  These  all  open 
upward,  allowing  the  air  to  pass  out,  but  preventing  its  return. 

249.  Operation. — When  the  piston  is  depressed,  the  air  below 
it,  by  its  increased  tension,  presses  down  the  first  valve,  and  opens 
the  second,  and  escapes  into  the  upper  part  of  the  barrel.     When 
the  piston  is  raised,  the  air  above  it  cannot  return,  but  is  pressed 
through  the  third  valve  into  the  open  air  ;  while  the  air  in  the  re- 
ceiver and  pipe,  by  its  tension,  opens  the  first  valve,  and  diffuses 
itself  equally  through  the  receiver  and  barrel.     Another  descent 
and  ascent  only  repeat  the  same  process  ; 

and  thus,  by  a  succession  of  strokes,  the 
air  is  nearly  all  removed. 

The  exhaustion  can  be  made  more 
complete  if  the  first  and  second  valves 
are  opened,  by  the  action  of  the  piston 
and  rod,  rather  than  by  the  tension  of 
the  air.  This  method  is  illustrated  by 
Fig.  167,  a  section  of  the  barrel  and  pis- 
ton. The  first  and  second  valves,  as 
shown  in  the  figure,  are  conical  or  pup- 
pet valves,  fitting  into  conical  sockets. 
The  first  has  a  long  stem  attached,  which 
passes  through  the  piston  air-tight,  and 
is  pulled  up  by  it  a  little  way,  till  it  is 
arrested  by  striking  the  top  of  the  barrel. 
The  second  valve  is  a  conical  frustum  on 
the  end  of  the  piston-rod.  When  the 
rod  is  raised,  it  shuts  the  valve  before 

moving  the  piston  ;  when  it  begins  to  descend,  it  opens  the  valve 
again  before  giving  motion  to  the  piston.  The  first  valve  is  shut 
by  a  lever,  which  the  piston  strikes  at  the  moment  of  its  reaching 
the  top.  The  oil  which  is  likely  to  be  pressed  through  the  third 
valve  is  drained  off  by  the  pipe  (on  the  right  in  both  figures)  into 
a  cup  below  the  pump. 

250.  Rate  of  Exhaustion.— The  quantity  removed,  by  suc- 
cessive strokes,  and  also  the  quantity  remaining  in  the  receiver, 
diminishes  in  the  same  geometrical  ratio.     For,  of  the  air  occupy- 


166 


PNEUMATICS. 


FIG.  168. 


ing  the  barrel  and  receiver,  a  barrel-full  is  removed  at  each  stroke, 
and  a  receiver-full  is  left.  If,  for  example,  the  receiver  is  three 
times  as  large  as  the  barrel,  the  air  occupies  four  parts  before  the 
descent  of  the  piston ;  and  by  the  first  stroke  one-fourth  is  re- 
moved, and  three-fourths  are  left.  By  the  next  stroke,  three- 
fourths  as  much  will  be  removed  as  before  (J  of  J,  instead  of  J  of 
the  whole),  and  so  on  continually.  The  quantity  left  obviously 
diminishes  also  in  the  rame  ratio  of  three-fourths.  In  general,  if 
b  expresses  the  capacity  of  the  barrel,  and  r  that  of  the  receiver 

r 
and  connecting-pipe,  the  ratio  of  each  descending  series  is-y . 

With  a  given  barrel,  the  rate  of  exhaustion  is  obviously  more  rapid 
as  the  receiver  is  smaller.     If  the  two  were  equal,  ten  strokes  would 
rarefy  the  air  more  than  a  thousand  times.     For  (£)10  =  3-^. 
As  a  term  of  this  series  can  never  reach  zero,  a  complete  ex- 
haustion can  never  be  effected  by  the  air-pump ; 
but  in  the  best  condition  of  a  well-made  pump, 
it  is  not  easy  to  discover  by  the  gauge  that  the 
vacuum  is  not  perfect. 

251.  Sprengel's  Pump.— This  apparatus  is 
too  slow  in  its  action  for  ordinary  lecture  illustra- 
tion, but  gives  a  much  better  vacuum  than  any 
piston  pump.    The  length  a  b  (Fig.  168)  must  be 
more  than  30  inches,  and  the  diameter 
of  the  tube  should  be  quite  small,  about    FlcL16[ 
3*3-  inch.     Mercury  from  the  funnel  F 
falls  down  the  tube  a  b  in  drops,  which 
carry  air  before  them  from  the  receiver, 
which  is   connected   with    the  exhaust 
branch  B  by  suitable  tubing. 

252.  The  Air  Condenser.— While 
the  air-pump  shows  the  tendency  of  air 
to  dilate  indefinitely,  as  the  compressing 
force  is  removed,  another  useful  instru- 
ment, the  condenser,  exhibits  the  in- 
definite compressibility  of  air.  Like 
the  pump,  it  consists  of  a  barrel  and 
piston,  but  its  valves,  one  in  the  piston 
and  one  at  the  bottom  of  the  barrel, 
open  downward.  Fig.  169  shows  the 
exterior  of  the  instrument.  If  it  be  screwed  upon  the  top  of  a 
strong  receiver  (Fig.  170),  with  a  stop-cock  connecting  them,  air 
may  be  forced  in,  and  then  secured  by  shutting  the  stop-cock. 


AIR    CONDENSER. 


167 


FIG.  170. 


When  the  piston  is  depressed,  its  own  valve  is  shut  by  the  in- 
creased tension  of  the  air  beneath  it,  and  the  lower 
one  opened  by  the  same  force.  When  the  piston 
is  raised,  the  lower  valve  is  kept  shut  by  the  con- 
densed air  in  the  receiver,  and  that  of  the  piston 
is  opened  by  the  weight  of  the  outer  air,  which 
thus  gets  admission  below  the  piston. 

The  quantity  of  air  in  the  receiver  increases  at 
each  stroke  in  an  arithmetical  ratio,  because  the 
same  quantity,  a  barrel- full  of  common  air,  is  added 
every  time  the  piston  is  depressed.  A  small 
Mariotte's  tube  is  attached  to  the  receiver,  to  show  how  many 
atmospheres  have  been  admitted. 

253.  Experiments  with  the  Air  Condenser.— If  the  re- 
ceiver be  partly  filled  with  water,  and  a  pipe  from  the  stop-cock 
extend  into  it,  then  when  the  condenser  has  been  used  and  re- 
moved, and  the  stop-cock  opened,  a  jet  of  water  will  be  thrown  to 
a  height  corresponding  to  the  tension  of  the  inclosed  air.  A  gas- 
bag being  placed  in  the  condenser,  then  filled  and  shut,  will  be- 
come flaccid  when  the  air  around  it  is  compressed.  A  thin  glass 
bottle,  sealed,  will  be  crushed  by  the  same  force.  By  these  and 
other  experiments  may  be  shown  the  effects  of  increased  tension. 

S54.  The  Bellows.— The  simple  or  hand-bellows  consists  of 
two  boards  or  lids  hinged  together,  and  having  a  flexible  leather 
round  the  edges,  and  a  tapering  tube  through  which  the  air  is 
driven  out.  In  the  lower  board  there  is  a  hole  with  a  valve  lying 
on  it,  which  can  open  inward.  On  separating  the  lids,  the  air  by 
its  pressure  instantly  lifts  the  valve  and  fills  the  space  between 
them  ;  but  when  they  are  pressed  together,  the  valve  shuts,  and 
the  air  is  compelled  to 
escape  through  the 
pipe.  The  stream  is 
intermittent,  passing 
out  only  when  pressure 
is  applied. 

The  compound  bel- 
loivs,  used  for  forges 
where  a  constant  stream 
is  needed,  are  made 
with  two  compartments. 
The  partition,  C  T 
(Fig.  171)  is  fixed,  and 
has  in  it  a  valve  V  opening  upward.  The  lower  lid  has  also  a, 
valve  V  opening  upward,  and  the  upper  one  is  loaded  with  weights. 


FIG.  171. 


168 


PNEUMATICS. 


The  pipe  T  is  connected  with  the  upper  compartment.  As  the 
lower  lid  is  raised  by  the  rod  A  B,  which  is  worked  by  the  lever 
E  B,  the  air  in  the  lower  part  is  crowded  through  V'  into  the 
upper  part,  whence  it  is  by  the  weights  pressed  through  the 
pipe  T  in  a  constant  stream.  When  the  lower  lid  falls,  the 
air  enters  the  lower  compartment  by  the  valve  V. 


FIG.  172. 


255.  The  Siphon.— If  a  bent  tube  ABC  (Fig.  172)  be 
filled,  and  one  end  immersed  in  a  vessel 

of  water,  the  liquid  will  be  discharged 
through  the  tube  so  long  as  the  outer  end 
is  lower  than  the  level  in  the  vessel. 
Such  a  tube  is  called  a  siphon,  and  is 
much  used  for  removing  a  liquid  from 
the  top  of  a  reservoir  without  disturbing 
the  lower  part.  The  height  of  the  bend  B 
above  the  fluid  level  must  be  less  than  34 
feet  for  water,  and  less  than  30  inches 
for  mercury.  The  reasons  for  the  motion 
of  the  water  are,  that  the  atmosphere  is 
able  to  sustain  a  column  higher  than  E  B, 
and  that  C  B  is  longer  than  E  B.  The 
two  pressures  on  the  highest  cross-section 
B  of  the  tube  are  unequal. 

For  the  pressure  at  B  towards  the 
right  is  equal  to  the  atmospheric  pressure, 
which  call  #,  minus  the  weight  of  the 
column  E  B,  which  call  b  ;  or  P  =  a  —  b. 
The  pressure  towards  the  left  at  B  is  equal 
to  a  minus  the  weight  of  the  column  C  B, 
greater  than  E  B,  and  this  weight  we  may 

call  b  +  c ;  or  P'  =  a  —  (b  +  c).    The  difference  of  these  pressures 
will  determine  the  motion  at  B. 

P  _  p'  —  (a  _  j)  _  ja  _  (j  +  c)  j  =  Cy 

and  this  excess  of  pressure  c  causes  a  flow  in  the  direction  E  B  C. 

The  excess  of  pressure  at  any  other  point  of  the  siphon  might 
have  been  discussed  in  the  same  general  way.  In  no  case  will 
water  flow  if  the  short  arm  exceeds  34  feet  in  length,  and  practi- 
cally it  must  be  less  than  this. 

If  the  tube  is  small,  it  may  be  filled  by  suction,  after  the  end  A 
is  immersed.  If  it  is  large,  it  may  be  inverted  and  filled,  and 
then  secured  by  stop-cocks,  till  the  end  is  beneath  the  water. 

256.  Siphon  Fountain.— In  order  that  the  flow  may  be 
maintained,  it  is  not  necessary  that  the  tube  should  contain  noth- 


SIPHON    FOUNTAIN 


169 


FIG   173. 


ing  but  liquid.     Air  may  collect  in  large  quantities  at  the  highest 
point  and  still  not  wholly  stop  the  action. 

Into  a  flask  F  (Fig.  173)  fit  an  air-tight  cork,  through  which 

pass  two  tubes,  one  #  entering  several  inches  into 

the  flask  and  terminating  in  a  fine  jet  a,  and  the 

other  d  c  ending  at  the  cork.     Through  the  tube  d  c 

pour  water  till  the  flask  is  filled  to  the  jet  a,  when 

inverted  as  in  the  figure.     Place  the  end  of  the  tube 

a  b  in  a  beaker  of  water  H,  and 

let  the  end  of  a  rubber  tube  lead  FIG.  174. 

from  d  to  a,  pail  upon  the  floor. 

The  water  in  the  flask  will  flow 

out  through  the  tube  c  d,  and  when 

the  tension  of  the  air  in  F  has  been 

sufficiently  lowered,  the  pressure 

of  the  atmosphere  upon  the  water 

in  the  beaker  H  will  force  it  up 

the  tube  b  a  and  out  through  the 

jet.     The  action  will  continue,  as 

in  any  other  form  of  siphon,  so 

long  as  water  is  supplied  to  the 

short  arm. 

257.  The  Suction  Pump.— 

The  section  (Fig.  174)  exhibits  the 

construction  of  the  common  suc- 
tion pump.    By  means  of  a  lever, 

the  piston   P  is  moved  up   and 

down  in  the  tube  A  V.     In  the 

piston  is  a  valve  opening  upward, 

and  at  the  top  of  the  pipe   V  C 

is  another  valve,  shown  at  V, 
also  opening  upward.  The  latter  valve  must 
be  at  a  less  height  than  34  feet  above  the 
water  G,  the  practical  limit  being  about  29 
feet,  depending  somewhat  upon  the  weight  of 
the  valves.  When  the  piston  P  is  raised,  its  valve  is  kept  shut 
by  the  pressure  of  the  atmosphere  above.  The  air  below  the 
piston  in  the  barrel  A  V  is  rarefied  and  presses  less  and  less  upon 
the  valve  V  until  at  last  its  tension,  together  with  the  weight  of 
the  valve,  is  less  than  the  tension  of  the  air  in  the  pipe  V  C  and 
the  valve  opens,  the  air  passing  through  from  below.  Now  the  ten- 
sion of  the  air  in  V  C  being  less  than  that  of  the  atmosphere,  a 
column  of  water  will  be  forced  up  the  pipe  to  a  height  such  that 
the  tension  of  the  air  in  the  pipe  together  with  the  weight  of 
the  column  of  water  shall  equal  the  pressure  of  the  external  air. 


170  PNEUMATICS. 

When  the  upward  motion  of  P  ceases,  the  valve  F  closes  by  its 
own  weight.  When  P  descends,  on  the  return  stroke,  the  air  be- 
tween it  and  V  is  compressed  till  its  tension  is  greater  than  that 
of  the  atmosphere  and  the  weight  of  the  valve  combined,  when 
the  valve  in  P  is  raised  and  the  compressed  air  escapes.  The  pis- 
ton being  raised  again,  the  water  rises  still  higher,  till  at  length 
it  passes  through  the  valve,  and  the  piston  dips  into  it ;  after 
this  the  water  above  P  is  lifted  to  the  discharge  spout  3,  while 
that  below  P  is  forced  to  follow  the  piston  in  its  upward  motion 
by  the  pressure  of  the  atmosphere,  as  before. 

258.  Calculation  of  the  Force. — For  simplicity  the  weight 
of  piston,  rod,  and  valve,  and  the  resistances  of  friction  will  be 
neglected.  Call  the  atmospheric  pressure  15  Ibs.  per  square  inch. 
Suppose  the  water  to  have  been  raised  to  the  point  H,  and  call 
the  downward  pressure  of  the  column  H  C,  m  Ibs.  per  square  inch, 
and  the  tension  of  the  rarefied  air  in  the  pipe  r  Ibs.  per  square 
inch.  Call  the  area  of  the  piston  Q  square  inches.  Now  the  tension 
of  the  inclosed  air  plus  the  weight  of  the  water  column  equals  the 
atmospheric  pressure,  or  r  -f  m  =  15  Ibs.,  whence  r  =  15  —  m. 

The  total  pressure  upon  the  top  of  the  piston  is  Q  x  15  Ibs* 
Upon  the  lower  side  of  the  piston  the  pressure  is  Q  x  ribs,  or  Q 
(15  —  m)  ;  the  difference  of  these  is  to  be  overcome  by  the  lifting 
force,  F  =  Q  x  15  Ibs.  —  Q  (15  —  m)  Ibs.  =  Q  x  m  Ibs.,  that  is 
to  say,  the  force  required  is  equal  to  the  weight  of  a  column  of 
water  whose  cross-section  equals  the  area  of  the  piston  and  whose 
height  is  H  C. 

Suppose  the  water  to  be  above  the  piston,  at  A.  Call  the 
pressure  downward  of  the  column  A  P,  m'  Ibs.  per  square  inch, 
and  the  pressure  downward  of  the  column  P  C,  n'  Ibs.  per  square 
inch.  The  pressure  upon  the  upper  side  of  P  is  Q  x  (15  -f-  m') 
Ibs.,  and  the  pressure  upon  the  lower  side  is  Q  x  (15  —  n')  Ibs. 
according  to  the  principle  of  transmitted  pressure  (Art.  173). 
The  force  required  is  the  difference  of  these  two  pressures ;  whence 
F  =  Q  x  (15  +  m')  Ibs.  —  Q  x  (15  —  n')  Ibs.  =  Q  x  (m'  +  »') 
Ibs.,  or  is  equal  to  the  weight  of  a  column  whose  cross-section 
equals  the  area  of  P  and  height  A  C,  as  in  the  previous  case. 
From  this  investigation  we  learn  that  only  the  area  of  the  piston 
and  height  of  water  in  the  pump  above  the  surface  of  the  cistern 
need  be  considered,  the  diameter  of  the  pipe  V  C  not  enter- 
ing the  calculation.  If  d  =  the  diameter  of  the  piston  in  deci- 
mals of  a  foot,  then  £  n  d2  =  its  area ;  J-  TT  d2  h  =  the  cubic  feet 
of  water,  li  being  the  height  in  feet ;  and  ±  TT  d?  h  x  62.5  =  the 
pounds  to  be  lifted.  The  atmosphere  has  acted  simply  to  trans- 
mit force,  and  has  not  lessened  the  work  in  any  way. 


THE     FIRE-ENGINE. 


171 


259. 

FIG. 


The 
175. 


A 


Forcing  Pump. — The  piston  of  the  forcing 
pump  (Fig.  175)  is  solid,  and  the  upper  valve 
V  opens  into  the  side  pipe  VS.  In  the 
ascent  of  the  piston,  the  water  is  raised  as  in 
the  suction  pump ;  but  in  its  descent,  a  force 
must  be  applied  to  press  the  water  which  is 
above  V  into  the  side  pipe  through  V. 

Let  the  height  P  C  =  h,  and  the  height 
B  A,  above  the  level  of  the  piston  =  h'.  The 
force  expended  at  any  instant  during  the  up- 
ward motion  of  the  piston  is  J  TT  d2  li  x  62.5 
Ibs.,  and  as  h  is  greatest  at  the  end  of  the  up- 
ward stroke  this  force  is  increasing.  On  the 
downward  stroke  the  force  is  J  rr  d2  h'  x  62.5 
Ibs.,  since  the  column  P  V  balances  the  column 
B  V,  leaving  only  B  A  =  h'  to  act ;  as  B  A 
is  greatest  at  the  end  of  the  down  stroke  this 
force  is  also  increasing. 

The  piston  is  only  one  of  many  contriv- 
ances for  producing  rarefaction  of  air  in  a 
pump-tube  ;  but  since  it  is  the  most  simple  and 
most  easily  kept  in  repair,  the  piston-pump  is 
generally  preferred  to  any  other. 

260.  The  Fire-Engine.— This  machine 
generally  consists  of  one  or  more  forcing 
~  pumps,  with  a  regulating  air-vessel,  though 
the  arrangement  of  parts  is  exceedingly  varied. 
Fig.  176  will  illustrate  the  princi- 
ples of  its  construction.  As  the 
piston,  P,  ascends,  the  water  is 
raised  through  the  valve,  V,  by  at- 
mospheric pressure.  As  P  descends, 
the  water  is  driven  through  Pinto 
the  air-vessel,  M,  whence  by  the 
condensed  air  it  is  forced  out  with- 
out interruption  through  the  hose- 
pipe, L.  The  piston  P'  operates 
in  the  same  way  by  alternate  move- 
ments. The  piston-rods  are  at- 
tached to  a  lever  (not  represented), 
to  which  the  strength  of  several  men 
can  be  applied  at  once  by  means  of  hand-bars  called  brakes. 

The  air-vessel  may  be  attached  to  any  kind  of  pump,  when- 
ever it  is  desired  to  render  the  stream  constant. 


FIG.  176. 


_^_...:.'   •^-•^  f,.,~L^ 


172 


PNEUMATICS. 


261.  Hero's  Fountain. — The  condensation  in  the  air- 
vessel,  from  which  water  is  discharged,  may  be  produced  by  the 
weight  of  a  column  of  water.  An  illustration  is  seen  in  Hero's 
fountain,  Fig.  177.  A  vertical  column  of  water  from  the  vessel,  A, 
presses  into  the  air-vessel,  B,  and  condenses  the  air  more  or  less, 
according  to  the  height  of  A  B.  From  the  top  of  this  vessel  an 
air-tube  conveys  the  force  of  the  compressed  air  to  a  second  air- 
vessel,  0}  which  is  nearly  full  of  water,  and  has  a  jet-pipe  rising 
from  it.  Since  the  tension  of  air  in  C  is  equal  to  that  in  B,  a 
jet  will  be  raised  which,  if  unobstructed,  would  be  equal  in  height 
to  the  compressing  column,  A  B. 

This  plan  has  been  employed  to  raise  water  from  a  mine  in 
Hungary,  and  hence  called  "  the  Hungarian  machine." 

The  principle  of  its  application  for  this  purpose  may  be  under- 
stood from  the  annexed  diagram. 


FIG.  177. 


PIG.  178. 


Let  A  represent  a  reservoir,  or  water  supply,  situated  on  high 
ground  at  an  elevation  above  the  mouth  of  the  shaft  greater  than 
the  depth  of  shaft  to  be  drained.  From  this  reservoir  a  pipe  D 
(Fig.  178)  passes  to  the  bottom  of  a  large  and  strong  air  chamber 


MANOMETERS. 


173 


B.  From  the  top  of  the  air  chamber  a  pipe  E  passes  to  the  top 
of  a  much  smaller  chamber  C,  at  the  bottom  of  the  shaft,  from 
the  bottom  of  which  passes  the  discharge  pipe  F,  having  a  valve  at 
v.  Suppose  the  necessary  valves  to  be  supplied.  Let  all  pipes 
and  both  chambers  be  filled  with  air  only.  Open  the  valve  y, 
which  will  allow  water  from  the  mine  to  flow  into  C,  driving  out 
the  air  through  E  into  B  and  out  through  the  waste  pipe  x,  which 
must  also  be  open.  Now  close  y  and  x  and  open  D,  which  will 
permit  water  to  flow  from  A  into  B,  compressing  the  air  in  B, 
which  pressure  will  be  communicated  through  the  air-pipe  E  to 
the  surface  of  the  water  in  C,  driving  it  out  through  F.  When  B 
is  full,  or  nearly  full,  of  water,  close  Z>,  open  x  and  y,  and  thus 
allow  water  to  flow  into  C  and  out  of  B.  When  C  is  full  and  B 
is  empty,  repeat  the  action  as  at  first.  For  a  shaft  100  feet  deep, 
the  air  chamber  B  should  be  at  least  four  times  the  capacity  of  G. 
If  the  height  of  the  reservoir  A  above  the  mouth  of  the  shaft  be 
less  than  the  depth  of  the  shaft  to  be  drained,  the  water  must  be 
raised  by  successive  lifts. 

262.  Manometers. — These  are  instruments  for  measuring 
the  tension  of  gases  or  vapors.  The  open  manometer  or  "open 
mercurial  gauge,"  as  applied  to  the  steam  boiler,  consists  simply 
of  a  thick  glass  tube,  standing  vertical,  both  ends  open,  the  lower 
end  dipping  into  mercury  contained  in  a  closed  cistern  ;  a  pipe 
connects  the  space  above  the  mercury  in  the  cistern  with  the 
steam  space  in  the  boiler.  When  the  tension  of  the  steam  is 
equal  to  one  atmosphere  the  pressure  upon  the  mercury  in  the 
cistern  will  be  balanced  by  the  pressure  of  the  air,  transmitted 
through  the  open  upper  end  of  the  tube.  As  the  steam  pressure 
increases,  the  mercury  will  rise  in  the  tube,  at  a  pressure  of  two 
atmospheres  standing  at  about  30  inches,  at 
three  atmospheres  at  60  inches,  and  so  on.  The 
pressure  of  steam  is  always  given  as  so  many 
pounds  above  one  atmosphere  ;  a  boiler  carry- 
ing 30  Ibs.  of  steam,  really  has  45  Ibs.  internal 
pressure,  15  Ibs.  of  which  is  counterbalanced 
by  the  pressure  of  the  external  air. 

For  high  pressure  a  very  long  open  mercurial 
gauge  would  be  required  ;  in  such  cases  the 
closed  manometer,  or  closed  mercurial  gauge  may 
be  used.  This  differs  from  the  former  in  having 
the  glass  tube  A  B  closed  at  the  top,  as  repre- 
sented in  Fig.  179.  In  this  instrument  the  theo- 
retical graduation  is  determined  by  Mariotte's 
law,  in 


FIG.  179. 


the  following  manner: 


Having  closed 


174  PNEUMATICS. 

communication  with  the  boiler  D,  and  opened  communication 
with  the  atmosphere,  both  above  the  mercury  in  the  cistern  and 
in  the  tube,  the  level  of  the  mercury  will  be  the  same  in  both, 
and  the  tension  of  the  air  in  the  tube  will  be  one  atmosphere. 
The  tube  being  supposed  of  uniform  bore  throughout,  the  volume 
of  the  compressed  air  will  be  proportioned  to  its  height;  the 
tension  will  be  equal  to  that  of  the  steam  diminished  by  the 
weight  of  the  column  of  mercury.  Call  the  length  of  the  tube 
C  A  measured  from  the  level  of  the  mercury  in  the  cistern  I ;  the 
total  pressure  in  Ibs.  p,  and  the  height  in  inches  of  the  column 
of  mercury  C2  by  h.  The  tension  of  the  air  t  in  the  space 
A  2  =  1  —  h  is  given  by  the  equation 


From  Mariotte's  Law  we  have 

1  •  P  —  ^  ' ' 

Eeducing, 

£2  _  (2  p  4.  i)  h  =  (30  —  2p)l 


-  •    whence  h  =  ^~i  ±  |/(30  -  2p)  I  +  (^-H 

<v  \         /& 

Suppose  the  tube  to  be  40  inches  long,  and  let  it  be  required  to 
determine  the  graduation  for  pressure  of  45  Ibs.  In  this  case  we 
have  to  determine  h  from  the  values  I  =  40,  p  =  60,  which  sub- 
stituted above  give 


h  =  -  —^  -  -V   (30  -  120)  40  + 


Whence  h  =  27.1  inches. 

The  lower  sign  of  the  radical  in  the  value  of  h  is  used,  as  the 
upper  would  give  an  impossible  result.  As  the  uniformity  of  the 
bore  of  the  tube  can  not  be  assured,  graduation  by  actual  trial  is 
the  only  accurate  method.  Variations  caused  by  changes  of 
temperature  of  the  inclosed  air  must  be  corrected  by  tables  for 
the  purpose.  As  the  figures  crowd  together  near  the  top  of  the 
tube,  as  shown  in  the  diagram,  it  has  been  proposed  to  substitute 
Fi  180  a  taPering>  conical  tube,  for  the  cylindrical  tube, 

giving  it  such  proportions  as  to  practically  cor- 
rect these  inequalities. 

A  metallic  gauge,  called  from  the  inventor,  a 
Bourdon  Gauge,  or  some  modification  of  it,  is  in 
very  common  use.  It  consists  of  a  flattened 
tube,  bent  as  in  Fig.  180,  the  closed  end  A  be- 
ing connected  with  a  toothed  sector  C,  When 
steam  is  admitted  through  the  stop-cock  at  P. 


HEIGHT    O.F    THE    ATMOSPHERE. 


175 


FIG.  181. 


the  curved  tube  tends  to  straighten  as  the  pressure  rises,  and  the 
motion  of  the  end  A  in  the  direction  of  the  arrow  turns  the  sector 
C  about  its  axis  o,  and  by  the  teeth  gives  motion  to  the  pinion 
D,  which  carries  the  index.  These  gauges  are  graduated  by 
comparison  with  a  standard  mercurial  gauge. 

263.  Apparatus  for  Preserving  a  Constant  Level. — 
Let  A  B  (Fig.  181)  be  a  reser- 
voir which  supplies  a  liquid  to 
the  vessel  C  D ;  and  suppose  it 
is  desired  to  preserve  the  level 
at  the  point  C  in  the  vessel, 
while 'the  liquid  is  discharged 
from  it  irregularly  or  at  inter- 
vals. So  long  as  the  mouth  of 
the  pipe  E  is  submerged  in  the 
liquid  in  C  D,  no  air  can  enter 
the  reservoir  A  B,  and  hence  no 
liquid  can  flow  from  it ;  but 
when  the  liquid  is  drawn  from 
C  D  so  that  the  level  C  falls, 
air  will  bubble  up  through 
the  pipe  E,  displacing  'liquid  in 
A  B  till  the  end  of  the  pipe  E 
is  again  closed  ;  this  action  will 
be  repeated  as  often  as  the  level  in  C  D  falls  below  C.  The 
pipe  E  should  be  of  greater  cross-section  than  the  pipe  H,  or 
else  there  must  be  a  great  head  of  water  in  A  B,  so  that  E  may 
supply  liquid  faster  than  H  can  discharge  it. 


CHAPTEE    III. 

THE  ATMOSPHERE— ITS  HEIGHT,  AND  MOTIONS. 

264.  Virtual   Height   of  the  Atmosphere.— When  two 

fluid  columns  are  in  equilibrium  with  each  other,  their  heights 
are  inversely  as  their  specific  gravities  (Art.  194).  The  specific 
gravity  of  mercury  is  10464  times  that  of  the  air  at  the  ocean 
level.  Therefore,  if  the  air  had  the  same  density  in  all  parts, 
its  height  would  be  found  by  the  proportion, 

1  :  10464::  2. 5  :  26160  feet, 

which  is  almost  five  miles.  Hence,  the  quantity  of  the  entire  at- 
mosphere of  the  earth  is  pretty  correctly  conceived  of  when  we 
imagine  it  having  the  density  of  that  which  surrounds  us,  and 
reaching  to  the  height  of  five  miles. 


176  PNEUMATICS. 

265.  Decrease  of  Density. — But  the  atmosphere  is  very 
far  from  being  throughout  of  uniform  density.     The  great  cause 
of  inequality  is  the  decreasing  weight  of  superincumbent  air  at 
increasing  altitudes.    The  law  of  diminution  of  density,  arising 
from  this  cause,  is  the  following : 

77ie  densities  of  the  air  decrease  in  a  geometrical  as  the  altitudes 
increase  in  an  arithmetical  ratio.  For,  let  us  suppose  the  air  tc> 
be  divided  into  horizontal  strata  of  equal  thickness,  and  so  thin 
that  the  density  of  each  may  be  considered  as  uniform  throughout. 
Let  a  be  the  weight  of  the  whole  column  from  the  top  to  the 
earth,  b  the  weight  of  the  whole  column  above  the  lowest  stratum, 
c  that  of  the  whole  column  above  the  second,  &c.  Then  the 
weight  of  the  lowest  stratum  is  a  — <  #,  and  the  weight  of  the 
second  is  b  —  c,  &c.  Now  the  densities  of  these  strata,  and  there- 
fore their  weights  (since  they  are  of  equal  thickness),  are  as  the 
compressing  forces ;  or, 

a  —  bib  —  c  ::  1)  :  c\ 
:.  a  c  —  b  c  =  b2  —  b  c  ;  /.  a  c  =  W ; 

.'.  a  :  b  ::  b  \c\ 

in  the  same  way,  b  :  c  : :  c  :  d ; 

that  is,  the  weights  of  the  entire  columns,  from  the  successive 
strata  to  the  top  of  the  atmosphere,  form  a  geometrical  series ; 
therefore,  the  densities  of  the  successive  strata,  varying  as  the  com- 
pressing forces,  also  form  a  geometrical  series.  If,  therefore,  at  a 
certain  distance  from  the  earth,  the  air  is  twice  as  rare  as  at  the 
surface  of  the  earth,  at  twice  that  distance  it  will  be  four  times  as. 
rare,  at  three  times  that  distance  eight  times  as  rare,  &c. 

By  barometric  observations  at  different  altitudes,  it  is  found 
that  at  the  height  of  three  and  a  half  miles  above  the  earth  the  air 
is  one-half  as  dense  as  it  is  at  the  surface.  Hence,  making  an 
arithmetical  series,  with  3J  for  the  common  difference,  to  denote 
heights,  and  a  geometrical  series,  with  the  ratio  of  £,  to  denote 
densities,  we  have  the  following; 

Heights,    3J,  7,  10£,  14,  17|,  21,24J-,  28,  31J,  35. 
Densities,  $,  J,  £,  -fr,  ^,  ^  ^,  jfa,  yfy,  y^. 

According  to  this  law,  the  air,  at  the  height  of  35  miles,  is  at 
least  a  thousand  times  less  dense  than  at  the  surface  of  the  earth. 
It  has,  therefore,  a  thousand  times  less  weight  resting  upon  it ; 
in  other  words,  only  one-thousandth  part  of  the  air  exists  above 
that  height. 

266.  Actual  Height  of  the  Atmosphere.— The  foregoing 
law,  founded  on  that  of  Mariotte,  cannot,  however,  be  applicable 
except  to  moderate  distances.     If  it  were  strictly  true,  the  atmos- 
phere would  be  unlimited.    But  that  is  impossible  on  a  revolving 


MOTIONS    OF    THE    AIR.  177 

body,  since  the  centrifugal  force  must  at  some  distance  or  other 
equal  the  force  of  gravity,  and  thus  set  a  limit  to  the  atmosphere ; 
and  that  limit  in  the  case  of  the  earth  is  more  than  20,000  miles 
high.  The  actual  height  of  the  atmosphere  is  doubtless  far  below 
this  ;  for  there  can  be  none  above  the  point  where  the  repellency 
of  the  particles  is  less  than  their  weight ;  and  the  repellency  di- 
minishes just  as  fast  as  the  density,  while  the  weight  diminishes 
very  slowly.  The  highest  portions  concerned  in  reflecting  the 
sunlight  are  about  45  miles  above  the  earth.  But  there  is  reason 
to  believe  that  the  air  extends  much  above  that  height,  probably 
100  or  200  miles  from  the  earth. 

267.  The  Motions  of  the  Air. — The  air  is  never  at  rest. 
When  in  motion,  it  is  called  wind.    The  equilibrium  of  the  atmos- 
phere is  disturbed  by  the  unequal  heat  on  different  parts  of  the 
earth.     The  air  over  the  hotter  portions  becomes  lighter,  and  is 
therefore  pressed  upward  by  the  cooler  and  heavier  air  of  the  less 
heated  regions.     And  the  motions  thus  caused  are  modified  as  to 
direction  and  velocity  by  the  rotation  of  the  earth  on  its  axis. 

268.  The  Trade  Winds. — The  most  extensive  and  regular 
system  of  winds  on  the  earth  is  known  by  the  name  of  the  trade 
winds,  so  called  on  account  of  their  great  advantage  to  commerce. 
They  are  confined  to  a  belt  about  equal  in  width  to  the  torrid 
zone,  but  whose  limits  are  four  or  five  degrees  further  north  than 
the  tropics. 

In  the  northern  half  of  this  trade-wind  zone  the  wind  blows 
continually  from  the  northeast,  and  in  the  southern  half  from  the 
southeast.  As  these  currents  approach  each  other,  they  gradually 
become  more  nearly  parallel  to  the  equator,  while  between  them 
there  is  a  narrow  belt  of  calms,  irregular  winds,  and  abundant 
rains. 

The  oblique  directions  of  the  trade  winds  are  the  combined 
effects  of  the  heat  of  the  torrid  zone  and  the  rotation  of  the  earth. 
The  cold  air  of  the  northern  hemisphere  tends  to  flow  directly 
south,  and  crowd  up  the  hot  air  over  the  equator.  In  like  man- 
ner, the  cold  air  of  the  southern  hemisphere  tends  to  flow  directly 
northward.  So  that  if  the  earth  were  at  rest,  there  would  be  north 
winds  on  the  north  side  of  the  equator,  and  south  winds  on  the 
south  side.  But  the  earth  revolves  on  its  axis  from  west  to  east, 
and  the  air,  as  it  moves  from  a  higher  latitude  to  a  lower,  has  only 
so  much  eastward  motion  as  the  parallel  from  which  it  came- 
/Therefore,  since  it  really  has  a  less  motion  from  the  west  than 
those  regions  over  which  it  arrives,  it  has  relatively  a  motion  from 
the  east.  This  motion  from  the  east,  compounded  with  the  motion 
from  the  north  on  the  north  side  of  the  equator,  and  with  that 
12 


178  PNEUMATICS. 

from  the  south  on  the  south  side,  constitutes  the  northeast  and 
southeast  tradewinds. 

The  limits  of  this  system  move  a  few  degrees  to  the  north 
during  the  northern  summer,  and  to  the  south  during  the 
northern  winter,  but  very  much  less  than  might  be  expected 
from  the  changes  in  the  sun's  declination. 

In  certain  localities  within  the  tropics  the  wind,  owing  to  pecu- 
liar configurations  of  coast  and  elevations  of  the  interior,  changes 
its  direction  periodically,  blowing  six  months  from  one  point,  and 
six  months  from  a  point  nearly  opposite.  The  monsoons  of  south- 
ern India  are  the  most  remarkable  example. 

269.  The  Return  Currents. — The   air  which  is  pressed 
upward  over  the  torrid  zone  must  necessarily  flow  away  northward 
and  southward  towards  the  higher  latitudes,  to  restore  the  equi- 
librium.    Hence,  there  are  south  winds  in  the  upper  air  on  the 
north  side  of  the  equator,  and  north  winds  on  the  south  side.    But 
these  upper  currents  are  also  oblique  to  the  meridians,  because, 
having  the  easterly  motion  of  the  equator,  they  move  faster  than 
the  parallels  over  which  they  successively  arrive,  so  that  a  motion 
from   the  west  is  combined  with  the  others,  causing  southwest 
winds  in  the  northern  hemisphere,  and  northwest  in  the  southern. 
These  motions  of  the  upper  air  are  discovered  by  observations 
made  on  high  mountains,  and  in  balloons,  and  by  noticing  the 
highest  strata  of  clouds.     It  is  to  be  borne  in  mind  that  although 
the  atmosphere  is  more  than  100  miles  high,  yet  the  lower  half 
does  not  extend  beyond  three  and  a  half  miles  above  the  earth 
(Art.  265). 

270.  Circulation  Beyond  the  Trade  Winds.— The  upper 
part  of  the  air  which  flows  away  from  the  equator  cannot  wholly 
retain  its  altitude,  because  of  the  diminishing  space  on  the  suc- 
cessive parallels.     About  latitude  30°,  it  is  so  much  accumulated 
that  it  causes  a  sensible  increase  of  pressure  (Art.  243),  and  begins 
to  descend  to  the  earth.     It  is  probable  that  some  of  the  descend- 
ing air  still  retains  its  oblique  motion  towards  higher  latitudes 
(for  the  prevailing  winds  of  the  northern  temperate  zone  are  from 
the  southwest,  and  of  the  southern  temperate  zone  from  the  north- 
west), while  a  part  joins  with   the  lower  air  which   is  moving 
towards  the  equator.     Only  so  much  of  the  rising  equatorial  mass 
can  flow  back  to  the  polar  regions  as  is  needed  to  supply  the 
comparatively  small  area  within  them.     On  account  of  the  suc- 
cessive descent  of  the  air  returning  from  the  equator,  there  is 
much  less  distinctness  and  regularity  in  the  general  circulation 
outside  of  the  torrid  zone  than  within  it.     Besides  this,  various 
local  causes,  such  as  mountain  ranges,  sea-coasts,  and  ocean  cur- 


VENTILATORS.  17y 

rents,  clear  and  cloudy  skies,  &c.,  mingle  their  effects  with  the 
more  general  circulation,  and  modify  it  in  every  possible  way. 

271.  Land  and  Sea  Breezes, — These  are  limited  circula- 
tions over  adjoining  portions  of  land  and  water,  the  wind  blowing 
from  the  water  to  the  land  in  the  day  time,  and  in  the  contrary 
direction  by  night.     When  the  sun  begins  to  shine  each  day,  it 
heats  the  land  more  rapidly  than  the  water.     Hence  the  air  01^ 
the  land  becomes  warmer  and  lighter  than  that  on  the  water,  and 
the  surface  current  sets  toward  the  land.     By  night  the  flow  is  re- 
versed, because  the  land  cools  most  rapidly,  and  the  air  above  it 
becomes  heavier  than  that  over  the  water.     These  effects  are  more 
striking  and  more  regular  in  tropical  countries,  but  are  common 
in  nearly  all  latitudes. 

272.  A  Current  Through  a  Medium. — There  are  some 
phenomena  relating  to  currents  moving  through  a  fluid,  either  of 
the  same  or  a  different  kind,  which  belong  alike  to  hydraulics  and 
pneumatics ;  a  brief  account  of  these  is  presented  here. 

If  a  stream  is  driven  through  a  medium,  it  carries  along  the 
adjoining  particles  by  friction  or  adhesion.  The  experiment  of 
Venturi  illustrates  thi^  kind  of  action,  as  it  takes  place  between 
the  particles  of  water.  A  reservoir  filled  with  water  has  in  it  an 
inclined  plane  of  gentle  ascent,  whose  summit  just  reaches  the 
edge  of  the  reservoir.  A  stream  of  water  is  driven  up  this  plane 
with  force  sufficient  to  carry  it  over  the  top  ;  but  in  doing  so,  it 
takes  out  continually  some  part  of  the  water  of  the  reservoir,  and 
will  in  time  empty  it  to  the  level  of  the  lowest  part  of  the  stream. 
A  stream  of  air  through  air  produces  the  same  effect,  as  may  be 
shown  by  the  flame  of  a  lamp  near  the  stream  always  bending  to- 
ward it.  In  like  manner,  water  through  air  carries  air  with  it ; 
when  a  stream  of  water  is  poured  into  a  vessel  of  water,  air  is  car- 
ried down  in  bubbles  ;  and  cataracts  carry  down  much  air,  which 
as  it  rises  forms  a  mass  of  foam  on  the  surface.  The  strong  wind 
from  behind  a  high  waterfall  is  owing  to  the  condensation  of  air 
brought  down  by  the  back  side  of  the  sheet. 

273.  Ventilators. — If  the  stream  passes  across  the  end  of  an 
open  tube,  the  air  within  the  tube  will  be  taken  along  with  the 
stream  and  thus  a  partial  vacuum  formed,  and  a  current  estab- 
lished.    It  is  thus  that  the  wind  across  the  top  of  a  chimney  in- 
creases the  draught  within.     To  render  this  effect  more  uniformly 
successful,  by  preventing  the   wind  from  striking  the  interior 
edge   of  the  flue,  appendages,    called  ventilators,  are   attached 
to  the  chimney  top.     A  simple  one,  which  is  generally  effec- 
tual, consists  of  a  conical  frustum  surrounding  the  flue  as  in 


180  PNEUMATICS 

Fig.  182,  so  that  the  wind,  on  striking  the  oblique  surface,  is 
thrown  over  the  top  in  a  curve,  which  is 
convex  upward.  The  same  mechanical  con- 
trivance is  much  used  for  the  ventilation  of 
public  halls  and  the  holds  of  ships.  A 
horizontal  cover  may  be  supported  by  rods, 
at  the  height  of  a  few  inches,  to  prevent 
the  rain  from  entering. 


274.  A  Stream  Meeting  a  Sur- 
face.— Though  the  moving  fluid  may  be 
elastic,  yet,  when  it  meets  a  surface,  it 
tends  to  follow  it,  rather  than  to  rebound 
from  it.  This  effect  is  partly  due  to  adhe- 
sion, and  partly  to  the  resistance  of  the 
medium  in  which  the  stream  moves.  It  will  not  only  follow  a 
plane  or  concave  surface,  but  even  one  which  is  convex,  provided 
the  velocity  of  the  current  is  not  too  great,  or  the  curvature  too 
rapid.  A  stream  of  air,  blown  from  a  pipe  upon  a  plane  surface. 
Will  extinguish  the  flame  of  a  lamp  held  in  the  direction  of  the 
surface  beyond  its  edge,  while,  if  the  lamp  be  held  elsewhere  near 
the  stream,  the  flame  will  point  toward  the  stream,  according  to 
Art.  272.  Hence,  snow  is  blown  away  from  the  windward  side  of 
a  tight  fence,  and  from  around  trees. 

275.  Diminution  of  Pressure  on  a  Surface. — When  a 
stream  is  thus  moving  along  a  surface,  the  fluid  pressure  on  that 
surface  is  slightly  diminished.  This  is  proved  by  many  experi- 
ments. If  a  curved  vane  be  suspended  on  a  pivot,  and  a  stream 
of  air  be  directed  tangentially  along  the  surface,  it  will  move  to- 
ward the  stream,  and  may  be  made  to  revolve  rapidly  by  repeating '' 
the  blast  at  each  half  revolution.  What  is  frequently  called  the 
pneumatic  paradox  is  a  phenomenon  of  the  same  kind.  A  stream 
of  air  is  blown  through  the  centre  of  a  disk,  against  another  light 
disk,  which,  instead  of  being  blown  off,  is  forcibly  held  near  to  it 
by  the  means.  The  pressure  is  diminished  by  all  the  radial  streams 
along  the  surface  contiguous  to  the  other  disk,  and  the  full  pressure 
on  the  outside  preponderates.  Another  form  of  the  experiment  is 
to  blow  a  stream  of  air  through  the  bottom  of  a  hemispherical  cup, 
in  which  a  light  sphere  is  lying  loosely.  The  sphere  cannot  be 
blown  out,  but,  on  the  contrary,  is  held  in,  as  may  be  seen  by  in- 
verting the  cup,  while  the  blast  continues.  It  appears  to  be  for  a 
reason  of  the  same  sort  that  a  ball  or  a  ring  is  sustained  by  a  jet 
of  water.  It  lies  not  on  the  top,  but  on  the  side  of  the  jet,  which 
diminishes  the  pressure  on  that  side  of  the  ball,  so  that  the  air  on 
the  outside  keeps  it  in  contact.  The  tangential  force  of  the  jot 


VOKTJLCES.  181 

causes  the  body  to  revolve  with  rapidity.     A  ball  can  be  sustained 
a  few  inches  high  by  a  stream  of  air. 

276.  Vortices  where  the  Surface  Ends. — As  a  current 
reaches  the  termination  of  the  surface  along  which  it  was  flowing, 
a  vortex  or  whirl  is  likely  to  occur  in  the  surrounding  medium 
behind  the  edge  of  the  surface.     Vortices  are  formed  on  water, 
whose  flow  is  obstructed  by  rocks ;  and  often  when  the  obstruct- 
ing body  is  at  a  distance  below  the  surface,  the  whirl  which  is  es- 
tablished there  is  communicated  to  the  top,  so  that  the  vortex  is 
seen,  while  its  cause  is  out  of  sight.     There  is  a  depression  at  the 
centre,  caused  by  the  centrifugal  force  ;  and  if  the  rotation  is 
rapid,  a  spiral  tube  is  formed,  in  which  the  air  descends  to  great 
depths.     These  are  called  whirlpools.     In  a  similar  manner  whirls 
are  produced  in  the  air,  when  it  pours  off  from  a  surface.     The 
eddying  leaves  on  the  leeward  side  of  a  building  in  a  windy  day 
often  indicate  such  a  movement,  though  it  may  have  no  perma- 
nency, the  vortex  being  repeatedly  broken  up  and  reproduced. 

277.  Vortices  by   Currents  Meeting. — But  vortices  are 
also  formed  by  counteracting  currents  in  an  open  medium.   When 
an  aperture  is  made  in  the  middle  of  the  bottom  of  a  vessel,  as  the 
water  runs  toward  it,  the  filaments  encounter  each   other,  and 
usually,  though  not  invariably,  they  establish  a  rotary  motion, 
and  form  a  whirlpool.     Vortices  are  a  frequent  phenomenon  of 
the  atmosphere,  sometimes  only  a  few  feet  in  diameter,  in  other 
instances  some  rods  or  even  miles  in  width.     The  smaller  ones, 
occurring  over  land,  are  called   whirlwinds ;    over  water,  water- 
spouts.   They  probably  originate  in  currents  which  do  not  exactly 
oppose  each  other,  but  act  as  a  couple  of  forces,  tending  to  produce 
rotation  (Art.  56). 

The  burning  of  a  forest  sometimes  occasions  whirlwinds,  which 
are  borne  away  by  the  wind,  and  maintain  their  rotation  for  miles. 
As  the  pressure  in  the  centre  is  diminished  by  the  centrifugal 
force,  substances  heavier  than  air,  as  leaves  and  spray,  are  likely 
to  be  driven  up  in  the  axis,  and  floating  substances,  as  cloud,  will 
for  the  same  reason  descend.  The  rising  spray  and  the  descend- 
ing cloud  frequently  mark  the  progress  of  a  vortex  in  the  air,  as 
it  moves  over  a  lake  or  the  ocean.  Such  a  phenomenon  is  called 
a  water-spout. 


PART    IY, 

.ACOUSTICS. 


CHAPTEE    I. 

NATURE  AND  PROPAGATION  OF  SOUND. 

278.  Sound.  —  Vibrations.  —  The  impression  which  the 
mind  receives  through  the  organ  of  hearing  is  called  sound.  But 
the  same  word  is  constantly  used  to  signify  that  progressive  vibra- 
tory movement  in  a  medium  by  which  the  impression  is  produced, 
as  when  we  speak  of  the  velocity  of  sound. 

This  is  one  of  the  several  modes  of  motion  mentioned  in  Art.  4. 
The  vibrations  constituting  sound  are  comparatively  slow,  and 
are  often  perceived  by  sight  and  by  feeling  as  well  as  by  hearing. 
For  these  reasons,  the  true  nature  of  sound  is  investigated  with 
far  greater  ease  than  that  of  light,  electricity,  &c.  It  is  not  diffi- 
cult to  discover  that  vibrations  in  the  medium  about  us  are  essen- 
tial to  hearing ;  and  these  vibrations  are  always  traceable  to  the 
body  in  which  the  sound  originates.  A  body  becomes  a  source  of 
sound  by  producing  an  impulse  or  a  series  of  impulses  on  the  sur- 
rounding medium,  and  thus  throwing  the  medium  itself  into 
motion.  A  single  sudden  impulse  causes  a  noise,  with  very  little 
continuance;  an  irregular  and  rapid  succession  of  impulses  a 
crash,  or  roar,  or  continued  noise  of  some  kind ;  but  if  the  im- 
pulses are  rapid  and  perfectly  equidistant,  the  effect  is  a  musical 
sound.  In  most  cases  of  the  last  kind  the  impulses  are  vibrations 
of  the  body  itself  ;  and  whatever  affects  these  vibrations  is  found 
to  affect  the  sound  emanating  from  it ;  and  if  they  are  destroyed, 
the  sound  ceases. 

If  we  rub  a  moistened  finger  along  the  edge  of  a  tumbler 
nearly  full  of  water,  or  draw  a  bow  across  the  strings  of  a  viol,  we 
can  procure  sounds  which  remain  undiminished  in  intensity  as 
long  as  the  operation  by  which  they  are  excited  is  continued.  In 
both  cases  the  vibrations  are  visible ;  those  of  the  tumbler  are 
plainly  seen  as  crispations  on  the  water  to  which  they  are  commu- 


AIR    AS    A    MEDIUM    OF    SOUND.  183 

nicated  ;  the  string  appears  as  a  broad  shadowy  surface.  If  a  wire 
or  light  piece  of  metal  rests  against  a  bell  or  glass  receiver,  when 
ringing,  it  will  be  made  to  rattle.  If  sand  be  strewed  on  a  hori- 
zontal plate  while  a  bow  is  drawn  across  its  edge,  the  sand  will  be 
agitated,  and  dance  over  the  surface,  till  it  finds  certain  places 
where  vibrations  do  not  exist.  Near  an  organ-pipe  the  tremor 
of  the  air  is  perceptible,  and  pipes  of  the  largest  size  jar  the  seats 
and  walls  of  an  edifice.  Every  species  of  sound  may  be  traced  to 
impulses  or  vibrations  in  the  sounding  body. 

279.  Sonorous  Bodies. — Two  qualities  in  a  body  are  neces- 
sary, in  order  that  it  may  be  sonorous.     It  must  have   a  form 
favorable  for  vibratory  movements,   and  sufficient   strength   of 
elasticity. 

The  favorable  forms  are  in  general  rods  and  plates,  rather  than 
very  compact  masses,  like  spheres  and  cubes ;  because  the  particles 
of  the  former  are  more  free  to  receive  lateral  movements  than 
those  of  the  latter,  which  are  constrained  on  every  side.  But  even 
a  thin  lamina  may  have  a  form  which  allows  too  little  freedom  of 
motion,  such  as  a  spherical  shell,  in  which  the  parts  mutually  sup- 
port each  other.  If  the  shell  be  divided,  the  hemispheres  are  bell- 
shaped  and  very  sonorous. 

The  elasticity  of  some  materials  is  too  imperfect  for  continued 
vibration  ;  thus  lead,  in  whatever  form,  has  no  sonorous  quality. 
In  other  cases,  where  the  elasticity  is  nearly  perfect,  yet  it  is  a 
feeble  force,  and  hence  the  vibrations  are  slow  and  inaudible.  Thus 
india-rubber  is  quite  elastic,  but  its  force  is  feeble,  and  occasions 
but  little  sound. 

280.  Air  as  a  Medium  of  Sound. — There  must  not  only 
be  a  vibrating  body,  as  a  source  of  sound,  but  a  medium  for  its 
communication  to  the  organ  of  hearing.     The  ordinary  medium  is 
air.     Let  a  bell  mounted  with  a  hammer  and  mainspring,  so  as  to 
continue  ringing  for  several  minutes,  be  placed  on  a  thick  cushion 
under  the  receiver  of  an  air-pump.     The  cushion,  made  of  several 
thicknesses  of  woolen  cloth,  is  necessary  to  prevent  communica- 
tion through  the  metallic  parts  of  the  instrument.     As  the  pro- 
cess of  exhaustion  goes  on,  the  sound  of  the  bell  grows  fainter, 
and  at  length  ceases  entirely.   From  this  experiment  we  learn  that 
sound  cannot  be  propagated  through  a  vacant  space,  even  though 
it  be  only  an  inch  or  two  in  extent  ;  and  also  that  air  conveys 
sound  more  feebly  as  it  is  more  rare.     The  latter  is  proved  by  the 
faintness  of  sounds  on  the  tops  of  high  mountains.    Travelers 
among  the  Alps  often  observe  that  at  great  elevations  a  gun  can 
be  hoard  orlv  a  small  diVtanp<\    The  fact  that  meteoric  bodies  are 


184  ACOUSTICS. 

sometimes  heard  when  passing  over  at  the  height  of  40  or  50  miles 
does  not  conflict  with  the  above  statements  ;  for  the  velocity  of 
meteors  is  vastly  greater  than  any  other  velocities  which  occur 
within  the  earth's  atmosphere.  On  the  other  hand,  when  air  has 
more  than  the  natural  density,  it  conveys  sound  with  more  inten- 
sity, and  therefore  to  a  greater  distance.  In  a  diving-hell  sunk  to 
a  considerable  depth  a  whisper  is  painfully  loud. 

281.  Velocity  of  Sound  in  Air. — Sound  occupies  an  ap- 
preciable time  in  passing  through  air.     This  is  a  fact  of  common 
observation.     The  flash  of  a  distant  gun  is  seen  before  the  report 
is  heard.     Thunder  usually  follows  lightning  after  an  interval  of 
many  seconds  ;  but  if  the  electric  discharge  is  quite  near,  the 
lightning  and  thunder  are  almost  simultaneous.     If  a  person  is 
hammering  at  a  distance,  the  perceptions  of  the  blows  received  by 
the  eye  and  the  ear  do  not  generally  agree  with  each  other:  or  if 
in  any  case  they  do  agree,  it  will  be  observed  that  the  first  stroke 
seen  is  inaudible,  and  the  last  one  heard  is  invisible ;  for  it  re- 
quires just  the  time  between  two  strokes  for  the  sound  of  each  to 
reach  us. 

A  long  column  of  infantry,  marching  to  the  music  of  a  single 
band,  will  have  a  vertical  wave-like  motion,  since  each  rank  steps 
to  the  music,  and  a  given  beat  reaches  the  different  ranks  in  suc- 
ceeding periods  of  time. 

Many  careful  experiments  were  made  in  the  eighteenth  cen- 
tury to  determine  the  velocity  of  sound  ;  but  as  the  temperature 
was  not  recorded,  they  have  but  little  value.  During  the  present 
century,  the  velocity  has  been  determined  by  several  series  of  ob- 
servations in  different  countries,  and  all  reduced  for  temperature 
to  the  freezing-point.  The  agreement  between  them  is  very  close, 
and  the  mean  of  all  is  1090  feet  per  second  at  32°  F. 

282.  Velocity  as  Affected  by  the  Condition  of  the 
Air  and  the  Quality  of  the  Sound. — 

Temperature  affects  the  velocity  of  spund ;  the  latter  is  in- 
creased about  one  foot  (1.11  ft.)  for  each  degree  Fahrenheit  of 
rise  in  the  temperature.  Therefore,  in  most  New  England 
climates,  the  velocity  of  sound  varies  more  than  100  feet  during 
the  year  on  this  account.  Probably  the  celebrated  experiments  of 
Derham,  in  London,  1708,  who  made  the  velocity  1142  feet,  were 
performed  in  the  heat  of  summer. 

Wind  of  course  affects  the  velocity  of  sound  by  the  addition 
or  subtraction  of  its  own  velocity,  estimated  in  the  same  direc- 
tion, because  it  transfers  the  medium  itself  in  which  the  sound 
is  conveyed.  This  modification,  however,  is  only  slight,  for 


DIFFUSION    OF    SOUND.  185 

sound  moves  ten  times  faster  than  wind  in  the  most  violent  hur- 
ricane. 

But  other  changes  in  the  condition  of  the  air  produce  little 
or  no  effect.  Neither  pressure,  nor  moisture,  nor  any  change  of 
weather,  alters  the  velocity  of  sound,  though  they  may  affect  its 
intensity)  and  therefore  the  distance  at  which  it  can  be  heard. 
Falling  snow  and  rain  obstruct  sound,  but  do  not  retard  it. 

All  kinds  of  sound — the  firing  of  a  gun — the  blow  of  a  ham- 
mer— the  notes  of  a. musical  instrument,  or  of  the  voice,  however 
high  or  low,  loud  or  soft,  are  conveyed  at  the  same  rate.  That 
sounds  of  different  pitch  are  conveyed  with  the  same  velocity  was 
conclusively  proved  by  Biot,  in  Paris,  who  caused  several  airs  to 
be  played  on  a  flute  at  one  end  of  a  pipe  more  than  3000  feet  long, 
and  heard  the  same  at  the  other  end  distinctly,  and  without  the 
slightest  displacement  in  the  order  of  notes,  or  intervals  of  silence 
between  them. 

283.  The  Calculated  Velocity. — For  several  years  there 
was  a  large  unexplained  difference  between  the  calculated  velocity 
of  sound  and  the  actual  velocity  as  determined  by  experiment. 
While  the  latter  is,  as  already  stated,  1090  feet  per  second  at  the 
freezing-point,  calculation  gave  916  feet.     The  difference  was  at 
length  explained  by  La  Place,  who  ascertained  that  it  arises  from 
the  heat  developed  in  the  air  by  the  compression  which  it  under- 
goes. 

The  velocity  of  sound  is  directly  proportional  to  the  square 
root  of  the  elasticity  and  inversely  proportional  to  the  square  root 
of  the  density;  but  according  to  Mariotte's  law,  the  elasticity 
varies  as  the  density,  and  hence  the  velocity  in  air  is  independent 
of  the  density,  the  temperature  being  constant. 

But  it  is  a  well-known  fact  that  when  air  is  compressed,  a  part 
of  its  latent  heat  becomes  sensible,  and  raises  its  temperature.  If 
the  condensation  is  gradual,  the  heat  is  radiated  or  conducted  off, 
especially  if  in  contact  with  other  bodies  ;  but  the  heat  developed 
in  the  propagation  of  sound  has  little  opportunity  to  escape,  and, 
though  without  continuance,  it  augments  the  elasticity  of  the  air, 
so  as  to  add  174  feet  to  the  velocity  of  sound  in  it. 

284.  Diffusion  of  Sound. — Sound  produced  in  the  open  air 
tends  to  spread  equally  in  all  directions,  and  will  do  so  whenever 
the  original  impulses  are  alike  on  every  side.     But  this  is  rarely 
the  case.     In  firing  a  gun,  the  first  impulse  is  given  in  one  direc- 
tion, and  the  sound  will  have  more  intensity,  and  be  heard  further 
in  that  direction  than  in  others.     It  is  ascertained  by  experiment, 
that  a  person  speaking  in  the  open  air  can  be  equally  well  heard 
at  the  distance  of  100  feet  directly  before  him,  75  feet  on  the  right 


186 


ACOUSTICS. 


and  left,  and  30  feet  behind  him ;  and  therefore  an  audience,  in 
order  to  hear  to  the  best  advantage,  should  be  arranged  within 
limits  having  these  proportions.  But,  as  will  be  seen  hereafter, 
this  rule  is  not  applicable  to  the  interior  of  a  building. 

Sound  is  also  heard  in  certain  directions  with  more  intensity, 
and  therefore  to  a  greater  distance,  if  an  obstacle  prevents  its  dif- 
fusion in  other  directions.  On  one  side  of  an  extended  wall  sound 
is  heard  further  than  if  it  spread  on  both  sides ;  still  further,  in  an 
angle  between  two  walls  ;  and  to  the  greatest  distance  of  all,  when 
confined  on  four  sides,  and  limited  to  one  direction,  as  in  a  long 
tube.  The  reason  in  these  several  cases  is  obvious  ;  for  a  given 
force  can  produce  a  given  amount  of  motion ;  and  if  the  motion  is 
prevented  from  spreading  to  particles  in  some  directions,  it  will 
reach  more  distant  ones  in  those  directions  in  which  it  does  spread. 
Speaking-tubes  confine  the  movement  to  a  slender  column  of  air, 
and  therefore  convey  sound  to  great  distances,  and  are  on  this 
account  very  useful  in  transmitting  messages  and  orders  between 
remote  parts  of  manufacturing  edifices  and  public  houses. 

285.  Nature  of  Acoustic  Waves. — The  vibrations  of  a 
medium  in  the  transmission  of  sound  are  of  the  kind  called  longi- 
tudinal ;  that  is,  the  particles  vibrate  longitudinally  with  regard 
to  the  movement  of  the  sound ;  whereas,  in  water-waves,  the  par- 
ticle-motion is  partly  transverse  to  the  wave-motion  (Art.  221). 
If,  for  example,  sound  is  passing  from  A  to  B  (Fig.  183),  the 

FIG.  183. 


particles  just  about  A  are  (at  the  moment  represented)  in  a  state  of 
condensation ;  around  this  condensed  centre  is  a  rarefied  portion, 
then  a  condensed  portion,  &c. ,  as  marked  by  the  letters  r,  c,  r' ,  c', 
r",  &c.  From  r  to  c  the  particles  are  advancing  ;  so  likewise  from 
r'  to  c',  and  from  r"  to  c".  But  from  c  to  r',  from  c'  to  r",  &c., 
they  are  rebounding.  The  condensed  wave  near  B  has  advanced 
from  A,  and  others  have  followed  it  at  equal  intervals ;  and  be- 
tween these  waves  of  condensation  are  waves  of  rarefaction,  which 
in  like  manner  spread  outward  from  the  centre  A.  And  yet  no 
one  particle  has  any  other  motion  than  a  small  vibration  back  and 
forth  in  the  line,  near  its  original  place  of  rest. 


INTENSITY    OF    SOUND.  187 

The  distance  from  a  particle  which  has  completed  one  vibra- 
tion to  the  particle  next  in  order  which  is  just  commencing  its 
vibration  is  called  a  wave  length,  and  all  the  intermediate  particles, 
which  represent  every  possible  phase  of  vibration,  constitute  the 
wave ;  or  we  may  define  a  wave  length  to  be  the  distance  between 
any  two  particles  in  consecutive  like  phases.  The  amplitude  \& 
the  distance  through  which  a  particle  vibrates. 

In  water-waves  we  distinguish  carefully  between  the  motion  of 
the  wave  and  the  motion  of  the  water  which  forms  the  wave  ;  so 
here,  the  wave-motion  is  totally  different  from  the  motion  of  the 
air  itself.  The  wave,  i.  e.  the  state  of  condensation  and  subsequent 
rarefaction,  travels  swiftly  forward  ;  but  the  masses  of  air,  which 
suffer  these  condensations  and  rarefactions,  simply  tremble  in  the 
line  of  that  motion. 

286.  Intensity  of  Sound. — Since  the  motion  is  propagated 
in  all  directions  alike,  the  entire  system  of  waves  around  the  point 
where  sound  originates  consists  of  spherical  strata  of  air  alter- 
nately condensed  and  rarefied.  As  the  quantity  set  in  motion  in 
these  successive  layers  increases  with  the  square  of  the  distance, 
the  amount  of  motion  communicated  to  each  particle  must  di- 
minish in  the  same  ratio.  Hence,  the  intensity  of  sound  varies 
inversely  as  the  square  of  the  distance. 

Intensity  increases  with  the  amplitude  of  the  vibrations,  and 
is  proportional  to  the  square  of  the  amplitude,  or  what  is  the 
same  thing,  it  is  proportional  to  the  square  of  the  maximum 
velocity  of  the  particle  vibrating.  To  obtain  a  loud  tone  from  a 
piano  its  keys  must  be  struck  with  great  force,  thus  increasing 
the  amplitude  of  vibration  of  the  strings. 

Intensity  depends  upon  the  density  of  the  air  in  which  the 
sound  is  produced,  but  not  upon  that  of  the  air  through  which  it 
is  transmitted.  A  sound  which  could  be  heard  in  water  at  a  dis- 
tance of  23  feet  would  be  audible  in  air  at  only  10  feet.  The  re- 
port of  a  cannon,  fired  upon  a  mountain  side,  heard  by  a  person 
in  the  rare  air  of  the  summit,  would  have  the  same  intensity  as 
the  same  report  heard  in  the  valley  below ;  but  a  gun  fired  in  the 
rare  air  of  the  summit  might  not  be  heard  in  the  valley,  while  a 
report  in  the  valley  would  be  heard  distinctly  upon  the  summit, 
the  intensity  depending  upon  the  density  of  the  medium  in  which 
the  sound  is  produced,  as  stated  above. 

Intensity  is  modified  by  motion  of  the  air.  In  still  air  sound 
is  more  perfectly  transmitted  than  when  air  currents  exist. ~  In 
case  of  winds  sound  is  more  intense,  for  a  given  distance,  in  the 
direction  of  the  wind  than  in  the  contrary  direction. 

Sound  is  strengthened  by  sympathetic  vibrations   of  other 


188  ACOUSTICS. 

bodies  than  that  which  first  produced  the  pulses.  A  vibrating 
string  produces  a  sound  scarcely  audible ;  but  when  it  vibrates 
upon  a  sounding  box,  the  sympathetic  vibrations  of  the  latter  are 
communicated  to  the  air  and  a  loud  sound  results.  A  vibrating 
tuning  fork  held  in  the  hand  can  not  be  heard  ;  the  same  fork 
caused  to  vibrate  over  the  mouth  of  a  cylinder  closed  at  one  end, 
and  of  a  length  equal  to  one-fourth  of  the  wave  length  correspond- 
ing to  the  pitch  of  the  fork  will  give  a  very  loud  sound.  Savart's 
resonator  illustrates  this  fact  very  satisfactorily. 

A  ray  of  sound  is  any  one  of  the  radii  of  the  sphere  whose 
centre  is  the  source  of  sound.  The  vibratory  motion  is  propagated 
along  each  of  the  rays. 

287.  Other  Gaseous  Bodies,  as  Media  of  Sound.— Let 

a  spherical  receiver,  having  a  bell  suspended  in  it,  be  exhausted  of 
air,  till  the  bell  ceases  to  be  heard  ;  then  fill  it  with  any  gas  or 
vapor  instead  of  air,  and  the  bell  will  be  heard  again.  By  means 
of  an  organ-pipe  blown  by  different  gases,  it  can  be  learned  with 
what  velocity  sound  would  move  in  each  kind  of  gas  experimented 
upon,  because  the  pitch  of  a  given  pipe  depends  upon  the  velocity 
of  the  waves,  as  will  be  seen  hereafter. 

From  such  experiments  the  following  velocities  at  tempera- 
ture 32°  Fahrenheit  have  been  deduced  • 

Carbonic  acid,  856  ft.  per  second. 

Oxygen,  1040  ft.  per  second. 

Carbonic  oxide,  1106  ft.  per  second. 

Hydrogen,  4163  ft.  per  second. 

It  has  been  stated  (Art.  283)  that  the  velocity  in  gases  varies 
directly  as  the  square  root  of  the  elasticities  and  inversely  as  the 
square  root  of  the  densities  ;  hence  for  the  same  pressures  the  velo- 
cities should  be  inversely  proportional  to  the  square  root  of  the 
densities.  Oxygen  being  16  times  as  heavy  as  hydrogen,  the 
velocity  of  sound  in  the  latter  should  be  four  times  as  great  as  in 
the  former,  which  conclusion  id  confirmed  by  the  facts  given  above. 
Momentary  development  of  heat  by  compression  produces,  in  all 
gaseous  bodies,  the  effect  of  increasing  the  velocity  of  sound. 

288.  Liquids  as  Media. — Many  experimenters  have  deter- 
mined the  circumstances  of  the  propagation  of  sound  in  water. 
Franklin  found  that  a  person  with  his  head  under  water  could 
hear  the  sound  of  two  stones  struck  together  at  a  distance  of  more 
than  half  a  mile.     In  1826,  Colladon  made  many  careful  experi- 
ments in  the  water  of  Lake  Geneva.     The  results  of  these  and 
other  trials  are  principally  the  following  : 

1.  Sounds  produced  in  the  air  are  very  faintly  heard  by  a  per- 


SOLIDS    AS    MEDIA.  189 

SOD  ill  water,  though  quite  near ;  and  sounds  originating  under 
water  are  feebly  communicated  to  the  air  above,  and  in  positions 
somewhat  oblique  are  not  heard  at  all. 

2.  Sounds  are  conveyed  by  water  with  a  velocity  of  4700  feet 
per  second,  at  the  temperature  of  47°  F.,  which  is  more  than  four 
times  as  great  as  in  air.     The  calculated  and  the  observed  velocity 
of  sound  in  water  agree  so  nearly  with  each  other,  that  there 
appears  to  be  no  appreciable  effect  arising  from  heat  developed  by 
compression. 

Calculated  velocities  are  as  follows: 
Seine  water,  at    59°  F.  =  4174  ft.  per  second. 
"        "  86°  F.  =  5013         "         " 

140°  F.  —  5657 

Solution  of  calcic  chloride,  at  73.4°  F.  =  6493  ft.  per  second. 
Sulphuric  ether  at  32°  F.  —  3801  ft.  per  second. 
Hence  sound    travels  with  different    velocities  in   different 
liquids  ;  the  velocity  is  greater  in  the  liquid  of  greater  density  ; 
the  velocity  is  increased  by  increase  of  temperature. 

3.  Sounds  conveyed  in  water  to  a  distance,  lose  their  sonorous 
quality.     For  example,  the  ringing  of  a  bell  gives  a  succession  of 
short  sharp  strokes,  like  the  striking  together  of  two  knife-blades. 
The  musical  quality  of  the  sound  is  noticeable  only  within  600  or 
700  feet.     In  air,  it  is  well  known  that  the  contrary  takes  place ; 
the  blow  of  the  bell- tongue  is  heard  near  by,  but  the  continued 
musical  note  is  all  that,  affects  the  ear  at  a  distance. 

4.  Acoustic  shadows  are  formed ;   that  is,   sound  passes  the 
edges  of  solid  bodies  nearly  in  straight  lines,  and  does  not  turn 
around  them  except  in  a  very  slight  degree.     In   this  respect, 
sound  in  water  resembles  light  much  more  than  it  does  sound  in 
air. 

To  enable  the  experimenter  to  hear  distant  sounds  without 
placing  himself  under  water,  Colladon  pressed  down  a  cylindrical 
tin  tube,  closed  at  the  bottom,  thus  allowing  the  acoustic  pulses  in 
the  water  to  strike  perpendicularly  on  the  sides  of  the  tube.  In 
this  way,  the  faintest  sounds  were  brought  out  into  the  air.  It 
appears  to  be  true  of  sound  as  of  light,  that  it  cannot  pass  from  a 
denser  to  a  rarer  medium  at  large  angles  of  incidence,  but  suffers 
nearly  a  total  reflection. 

289.  Solids  as  Media. — Solid  bodies  of  high  elastic  energy 
are  the  most  perfect  media  of  sound  which  are  known.  An  iron 
rod — as,  for  instance,  a  lightning-rod — will  convey  a  feeble  sound 
from  one  extremity  to  the  other,  with  much  more  distinctness 
than  the  air.  If  the  ears  are  stopped,  and  one  end  of  a  long  wire 
is  held  between  the  teeth,  a  slight  scratch  or  blow  on  the  remote 


190 


ACOUSTICS 


end  will  sound  very  loud.  The  sound  in  this  case  travels  through 
the  wire  and  the  bones  of  the  head  to  the  organ  of  hearing.  The 
sound  of  earthquakes  and  volcanic  eruptions  is  transmitted  to 
great  distances  through  the  solid  earth.  By  laying  the  ear  to  the 
ground,  the  tramp  of  cavalry  may  be  heard  at  a  much  greater  dis- 
tance than  through  the  air. 

290.  Velocity  in  Solids.— Structure. — The  velocity  of 
sound  in  cast  iron  was  estimated  by  Biot  to  be  about  11000  feet 
per  second — ten  times  greater  than  in  air.  He  obtained  this  re- 
sult by  experiments  on  the  aqueduct  pipes  in  Paris.  A  blow  upon 
one  end  was  brought  to  an  observer  at  the  other  end,  3000  feet 
distant,  both  by  the  iron  and  also  by  the  air  within  it.  The  velo- 
city in  air  being  known,  and  the  difference  of  time  observed,  the 
velocity  in  iron  is  readily  calculated ;  thus,  suppose  the  temperature 
of  the  air  to  have  been  41°  R,  the  length  of  the  pipe  3300  feet, 

,          X  3300     3300 

and  the  observed  interval  of  time  2T^  seconds,  then  — — —  = 

11 00         x 

2TV  in  which  1100  is  the  velocity  of  sound  in  air  at  the  given 
temperature,  and  x  the  velocity  in  the  pipe :  from  this  we  get 
x  =  11000. 

The  following  table,  according  to  Wertheim,  is  taken  from 
Tyndall : 


NAMB  or  METAL. 

At  20°  C. 

At  100°  C. 

At  200°  C. 

Lead  

4030 

3951 

Gold  

5717 

5640 

5691 

Silver  

8553 

8658 

8127 

Copper     ...                . 

11666 

10802 

9890 

Iron  

16822 

17386 

15483 

Iron  wire  

16130 

16728 

Steel  wire  >-.  

16023 

16443 

As  a  rule  the  velocity  in  metals  decreases  with  rise  of  tempera- 
ture, but  iron  and  silver  are  shown  above  to  be  exceptions  to  this 
general  rule  between  the  limits  20°  C.  and  100°  C. 

In  one  important  particular  solids  differ  from  fluids,  namely, 
in  the  fixed  relations  of  the  particles  among  themselves.  These 
relations  are  usually  different  indifferent  directions  ;  hence,  sound 
is  likely  to  be  transmitted  more  perfectly  in  some  directions  through 
a  given  solid  than  in  others.  The  scratch  of  a  pin  at  one  end  of 
a  stick  of  timber  seems  loud  to  a  person  whose  ear  is  at  the  other 
end.  The  sound  is  heard  more  perfectly  in  the  direction  of  the 
grain  than  across  it.  In  crystallized  substances  it  is  unquestion- 


MIXED    MEDIA.  191 

nbly  true  that  the  vibrations  of  sound  move  with  different  speed 
and  with  different  intensity  in  the  line  of  the  axis,  and  in  a  line 
perpendicular  to  it. 

The  velocity  in  woods  along  the  fibre  is  from  about  11000  feet 
to  16000  feet ;  across  the  annual  rings  from  4500  feet  to  6000  ; 
across  the  fibre,  in  the  direction  of  the  rings  from  about  2500 
feet  to  4500  feet,  all  of  which  velocities  are  approximate  and  de- 
pend upon  the  wood  selected. 

291.  Mixed  Media.— In  all  the  foregoing  statements  it 
has  been  supposed  that  the  medium  was  homogeneous  ;  in  other 
words,  that  the  material,  its  density,  and  its  structure,  continue 
the  same,  or  nearly  the  same,  the  whole  distance  from  the  source 
of  sound  to  the  ear.  If  abrupt  changes  occur,  even  a  few  times, 
the  sound  is  exceedingly  obstructed  in  its  progress.  When  the 
receiver  is  set  over  the  bell  on  the  pump  plate,  the  sound  in  the 
room  is  very  much  weakened,  though  the  glass  may  not  be  one- 
eighth  of  an  inch  in  thickness,  and  is  an  excellent  conductor  of 
sound.  The  vibrations  of  the  internal  air  are  very  imperfectly 
communicated  to  the  glass,  and  those  received  by  the  glass  pass 
into  the  air  again  with  a  diminished  intensity.  If  a  glass  rod 
extended  the  whole  distance  from  the  bell  to  the  ear,  the  sound 
would  arrive  in  less  time,  and  with  more  loudness,  than  if  air  oc- 
cupied the  whole  extent.  For  a  like  reason,  walls,  buildings,  or 
other  intervening  bodies,  though  good  conductors  of  sound  them- 
selves, obstruct  the  progress  of  sound  in  the  air.  This  explains 
the  fact  mentioned  in  Art.  288,  that  sound  in  air  is  heard  faintly 
in  water,  and  vice  versa.  When  the  texture  of  a  substance  is 
loose,  having  many  alternations  of  material,  it  thereby  becomes 
unfit  for  transmitting  sound.  It  is  for  this  reason  that  the  bell- 
stand,  in  the  experiment  just  referred  to,  is  set  on  a  cushion  made 
of  several  thicknesses  of  loose  flannel,  that  it  may  prevent  the  vi- 
brations from  reaching  the  metallic  parts  of  the  pump.  The  waves 
of  sound,  in  attempting  to  make  their  way  through  such  a  sub- 
stance, continually  meet  with  new  surfaces,  and  are  reflected  in 
all  possible  directions,  by  which  means  they  are  broken  up  into  a 
multitude  of  crossing  and  interfering  waves,  and  are  mutually 
destroyed.  A  tumbler,  nearly  filled  with  water,  will  ring  clearly; 
but  if  filled  with  an  effervescing  liquid,  it  will  lose  all  its  sonorous 
quality,  for  the  same  cause  as  before.  The  alternate  surfaces  of  the 
liquid  and  gas,  in  the  foam,  confuse  the  waves,  and  deaden  the 
;sound. 


CHAPTER    II. 


FIG.  184. 


REFLECTION,  REFRACTION,  AND  INFLECTION  OF  SOUND. 

292.  Reflection  of  Sound. — Sound  is  reflected  from  surfaces 
in  accordance  with  the  com'mon  law  of  reflection  in  the  case  of 
elastic  bodies  ;  that  is, 

The  angle  of  incidence  equals  the  angle  of  reflection,  and  the 
two  angles  are  on  opposite  sides  of  the  perpendicular  to  the  reflecting 
surface. 

Suppose  sound  to  emanate  from  A  (Fig.  184),  and  meet  the 
plane  surface  B  D.  The  particles  of  air  in  the  ray  A  B  vibrate 
back  and  forth  in  that  line, 
and  those  contiguous  to  B 
will,  after  striking  the  sur- 
face, rebound  on  the  line 
B  G,  as  an  elastic  ball  would 
do  (Art.  101),  and  propa- 
gate their  motion  along 
that  line.  The  angle  of 
incidence  A  B  F equals  the 
angle  of  reflection  F  B  G9 
and  the  two  angles  are  on 
opposite  sides  of  FB,  which 
is  perpendicular  to  the  re- 
flecting surface  B  D.  If 
G  B  be  produced  back- 
ward, it  will  meet  the  per- 
pendicular A  E  at  C,  as 
far  behind  B  D  as  A  is  before  it.  In  like  manner,  every  ray  of 
sound  after  reflection  proceeds  as  if  from  C,  and  the  successive 
waves  are  situated  as  represented  by  the  dotted  lines  in  the  figure. 
From  the  point  E  the  reflection  is  directly  back  in  the  line  E  A. 

293.  Echoes. — When  sound  is  so  distinctly  reflected  from  a 
surface  that  it  seems  to  come  from  another  source,  it  is  called  an 
echo.    Broad  and  even  surfaces,  such  as  the  walls  of  buildings  and 
ledges  of  rock,  often  produce  this  effect.     According  to  the  law 
(Art.  292),  a  person  can  hear  the  echo  of  his  own  voice  only  by 
standing  in  a  line  which  is  perpendicular  to  the  echoing  surface. 
In  order  that  one  person  may  hear  the  echo  of  another's  voice, 


SIMPLE  AND  COMPLEX  ECHOES.       193 

they  must  place  themselves  in  lines  making  equal  angles  with  the 
perpendicular. 

The  interval  of  time  between  a  sound  and  its  echo  enables  one 
to  judge  of  the  distance  of  the  surface,  since  the  sound  must  pass 
over  it  twice.  Thus,  if  at  the  temperature  of  74°  the  echo  of  the 
speaker's  voice  reaches  him  in  two  seconds  after  its  utterance,  the 
distance  of  the  reflecting  body  is  about  1130  feet,  and  in  that  pro- 
portion for  other  intervals.  And  he  can  hear  a  distinct  echo  of 
as  many  syllables  as  he  can  pronounce  while  sound  travels  twice 
the  distance  between  himself  and  the  echoing  surface. 

The  ear  can  recognize  about  nine  successive  sounds  in  one 
second  ;  two  sounds  separated  by  less  than  one-ninth  of  a  second 
blend  and  produce  confusion;  therefore  the  distance  from  the 
speaker  to  the  reflecting  surface  at  temperature  32°  F.  must  not 
be  less  than  1^  -r-  2  =  60.5  ft.  in  order  that  an  echo  of  a  sharp 
sound  may  be  heard.  For  articulate  sounds  at  ordinary  tempera- 
tures the  distance  may  be  about  112.5  feet. 

294.  Simple  and  Complex  Echoes. — When  a  sound  is 
returned  by  one  surface,  the  echo  is  called  simple;  it  is  called 
complex  when  the  reflection  is  from  two  or  more  surfaces  at 
different  distances,  each  surface  giving  one  echo.  Thus,  a  cannon 
fired  in  a  mountainous  region  is  heard  for  a  long  time  echoed  on 
all  sides,  and  from  various  distances. 

A  complex  echo  may  also  be  produced  by  two  parallel  walls,  if 
the  hearer  and  the  source  of  sound  are  both  situated  between 
them.  The  firing  of  a  pistol  between  parallel  walls  a  few  hundred 
feet  apart  has  been  known  to  return  from  30  to  40  echoes  before 
they  became  too  faint  to  be  heard.  The  rolling  of  thunder  is  in 
part  the  effect  of  reverberation  between  the  earth  and  the  clouds. 
This  is  made  certain  by  the  observed  fact  that  the  report  of  a  can- 
non, which  in  a  level  country  and  under  a  clear  sky  is  sharp  and 
single,  becomes  in  a  cloudy  day  a  prolonged  roar,  mingled  with 
distant  and  repeated  echoes.  But  the  peculiar  inequalities  in  the 
reverberations  of  thunder  are  doubtless  due  in  part  to  the  irregu- 
larly crinkled  path  of  the  electric  spark.  A  discharge  of  light- 
ning occupies  so  short  a  time,  that  the  sound  may  be  considered 
as  starting  from  all  points  of  its  track  at  once.  But  that  track  is 
full  of  large  and  small  curves,  some  convex  and  some  concave  to 
the  ear,  and  at  a  great  variety  of  distance  ;  and  all  points  which 
are  at  equal  distances  would  be  heard  at  once.  Hence,  the  origi- 
nal sound  comes  to  the  hearer  with  great  irregularity,  loud  at  one 
instant  and  faint  at  another.  These  inequalities  are  prolonged 
and  intensified  by  the  echoes  which  take  place  between  the  clouds 
and  the  earth. 


194  ACOUSTICS. 

295.  Concentrated  Echoes.— The  divergence  of  sound  from 
a  plane  surface  continues  the  same  as  before,  that  is,  in  spherical 
waves,  whose  centre  is  at  the  same  distance  behind  the  plane  as 
the  real  source  is  in  front.     But  concave  surfaces  in  general  pro- 
duce a  concentrating  effect.     A  sound  originating  in  the  centre 
of  a  hollow  sphere  will  be  reflected  back  to  the  centre  from  every 
point  of  the  surface.    If  it  emanates  from  one  focus  of  an  ellipsoid, 
it  will,  after  reflection,  all  be  collected  at  the  other  focus.     So,  if 
two  concave  paraboloids  stand  facing  each  other,  with  their  axes 
coincident,  and  a  whisper  is  made  at  the  focus  of  one,  it  will  be 
plainly  heard  at  the  focus  of  the  other,  though  inaudible  at  all 
points  between.     In  the  last  case  the  sound  is  twice  reflected,  and 
passes  from  one  reflector  to  the  other  in  parallel  lines.     All  these 
effects  are  readily  proved  from  the  principle  that  the  angles  of  in- 
cidence and  reflection  are  equal. 

The  speaking-trumpet  and  the  ear-trumpet  have  been  supposed 
by  many  writers  to  owe  their  concentrating  power  to  multiplied 
reflections  from  the  inner  surface.  But  a  part  of  the  effect,  and 
sometimes  the  whole,  is  doubtless  due  to  the  accumulation  of  force 
in  one  direction,  by  preventing  lateral  diffusion,  till  the  intensity 
is  greatly  increased. 

Concave  surfaces  cause  all  the  curious  effects  of  what  are  called 
whispering  galleries,  such  as  the  dome  of  St.  Paul's,  in  London. 
In  many  of  these  instances,  however,  there  seems  to  be  a  con- 
tinued series  of  reflections  from  point  to  point  along  the  smooth 
concave  wall,  which  all  meet  simultaneously  (if  the  curves  are  of 
equal  length)  at  the  opposite  point  of  the  dome ;  for  the  whisperer 
places  his  mouth,  and  the  hearer  his  ear,  close  to  the  wall,  and  not 
in  a  focus  of  the  curve.  The  Ear  of  Dionysius  was  probably  a 
curved  wall  of  this  kind  in  the  dungeons  of  Syracuse.  It  is  said 
that  the  words,  and  even  the  whispers,  of  the  prisoners  were 
gathered  and  conveyed  along  a  hidden  tube  to  the  apartment  of 
the  tyrant.  The  sail  of  a  ship  when  spread,  and  made  concave  by 
the  breeze,  has  been  known  to  concentrate  and  render  audible  to 
the  sailors  the  sound  of  a  bell  100  miles  distant.  A  concave  shell 
held  to  the  ear  concentrates  such  sounds  as  may  be  floating  in  the 
air,  and  is  suggestive  of  the  murmur  of  the  ocean. 

296.  Resonance  of  Rooms.— If  a  rectangular  room  has 
smooth,  hard  walls,  and  is  unfurnished,  its  reverberations  will  be 
loud  and  long-continued.     Stamp  on  the  floor,  or  make  any  other 
sudden  noise,,  and  its  echoes  passing  back  and  forth  will  form  a 
prolonged  musical  note,  whose  pitch  will  be  lower  as  the 'apart- 
ment is  larger.     This  is  called  tlfe  resonance  of  the  room.     Now, 
let  furniture  be  placed  around  the  walls,  and  the  reverberations 


HALLS    FOR    PUBLIC     SPEAKING.  195 

will  be  weakened  and  less  prolonged.  Especially  will  this  be  the 
case  if  the  articles  be  of  the  softer  kinds,  and  have  irregular  sur- 
faces. Carpets,  curtains,  stuffed  seats,  tapestry,  and  articles  of 
dress  have  great  influence  in  destroying  the  resonance  of  a  room. 
The  appearance  of  an  apartment  is  not  more  changed  than  is  its 
resonance  by  furnishing  it  with  carpet  and  curtains.  The  blind, 
on  entering  a  strange  room,  can,  by  the  sound  of  the  first  step, 
judge  with  tolerable  accuracy  of  its  size  and  the  general  character 
of  its  furniture. 

The  reason  why  substances  of  loose  texture  do  not  reflect  sound 
well,  is  essentially  the  same  as  what  has  been  stated  (Art.  291)  for 
their  not  transmitting  well  ;  they  are  not  homogeneous — the  waves 
are  reflected  in  all  directions  by  successive  surfaces,  interfere  with 
each  other,  and  are  destroyed. 

297.  Halls  for  Public  Speaking.— In  large  rooms,  such  ag 
churches  and  lecturing  halls,  all  echoes  which  can  accompany  the 
voice  of  the  speaker  syllable  by  syllable,  are  useful  for  increasing 
the  volume  of  sound  ;  but  all  which  reach  the  hearers  sensibly 
later,  only  produce  confusion.  It  is  found  by  experiment  that  if 
a  sound  and  its  echo  reach  the  ear  within  one-sixteenth  of  a  second 
of  each  other,  they  seem  to  be  one.  Hence,  this  fraction  of  time 
is  called  the  limit  of  perceptibility.  Within  that  time  an  echo  can 
travel  about  70  feet  more  than  the  original  sound,  and  yet  appear 
to  coincide  with  it.  If  an  echoing  wall,  therefore,  is  within  35 
feet  of  the  speaker,  each  syllable  and  its  echo  will  reach  every  hearer 
within  the  limit  of  perceptibility.  The  distance  may,  however,  be 
increased  to  40  or  even  50  feet  without  injury,  especially  if  the 
utterance  is  not  rapid.  Walls  intended  to  aid  by  their  echoes 
should  be  smooth,  but  not  too  solid ;  plaster  on  lath  is  better  than 
plaster  on  brick  or  stone  ;  the  first  echo  is  louder,  and  the  rever- 
berations less.  Drapery  behind  the  speaker  deprives  him  of  the 
aid  of  just  so  much  echoing  surface.  A  lecturing  hall  is  improved 
by  causing  the  wall  behind  the  speaker  to  change  its  direction, 
on  the  right  and  left  of  the  platform,  at  a  very  obtuse  angle,  so 
as  to  exclude  the  rectangular  corners  from  the  room.  The  voice 
is  in  this  way  more  reinforced  by  reflection,  and  there  is  less 
resonance  arising  from  the  parallelism  of  opposite  walls.  Panel- 
ing, and  any  other  recesses  for  ornamental  purposes,  may  exist  in 
the  reflecting  walls  without  injury,  provided  they  are  not  curved. 
The  ceiling  should  not  be  so  high  that  the  reflection  from  it 
would  be  delayed  beyond  the  limit  of  perceptibility.  Concave 
surfaces,  such  as  domes,  vaults,  and  broad  niches,  should  be  care- 
fully avoided,  as  their  effect  generally  is  to  concentrate  all  the 
sounds  they  reflect.  An  equal  diffusion  of  sound  throughout  the 


196 


ACOUSTICS. 


apartment,  not  concentration  of  it  to  particular  points,  is  the  ob- 
ject to  be  sought  in  the  arrangement  of  its  parts. 

As  to  distant  parts  of  a  hall  for  public  speaking,  the  more  com- 
pletely all  echoes  from  them  can  be  destroyed,  the  more  favorable 
is  it  for  distinct  hearing.  It  is  indeed  true  that  if  a  hearer  is 
within  35  feet  of  a  wall,  however  remote  from  the  speaker,  he  will 
hear  a  syllable,  and  its  echo  from  that  wall,  as  one  sound  ;  but  to 
all  the  audience  at  greater  distances  from  the  same  wall,  the  echoes 
will  be  perceptibly  retarded,  and  fall  upon  subsequent  syllables, 
thus  destroying  distinctness.  The  distant  walls  should,  by  some 
means,  be  broken  up  into  small  portions,  presenting  surfaces  in 
different  directions.  A  gallery  may  aid  in  effecting  this  ;  and  the 
seats  of  the  gallery  and  of  the  lower  floor  may  rise  rapidly  one 
behind  another,  so  that  the  audience  will  receive  directly  much  of 
the  sound  which  would  otherwise  go  to  the  remote  wall,  and  be 
reflected.  Especially  should  no  large  and  distant  surfaces  be 
parallel  to  nearer  ones,  since  it  is  between  parallel  walls  that  pro- 
longed reverberation  occurs. 

298.  Refraction  of  Sound. — It  has  been  ascertained  by 
experiment  that  sound,  like  light,  may  be  refracted,  or  bent  out  of 
its  rectilinear  course  by  entering  a  substance  of  different  density. 
If  a  large  convex  lens  be  formed  of  carbonic  acid  gas,  by  inclosing 
it  in  a  sphere  of  thin  india-rubber,  a  feeble  sound,  like  the  ticking 
of  a  watch,  produced  on  one  side,  will  be  concentrated  to  a  focal 
point  on  the  other.     In  this  case,  the  several  diverging  rays  of 
sound  are  refracted  toward  each  other  on  entering  the  sphere,  and 
still  more  on  leaving  it,  so  that  they  are  converged  to  a  focus. 

299.  Inflection  of  Sound. — If  air-waves  are  allowed  to  pass 
through  an  opening  in  an  obstructing  wall,  they  are  not  entirely 
confined  within  the  radii  of  the  wave-system  produced  through 
the  opening,  but  spread  with  diminished  intensity  in  lateral  direc- 
tions.    The  particles  near 

the  edges  of  the  opening, 
as  B  and  C  (Fig.  185)  may 
be  considered  as  sources 
of  sound  ;  and  if  they  be 
made  centres  of  concentric 
spheres,  whose  radii  are 
equal  to  the  length  of  the 
wave,  B  b,  or  C  c,  and  its 
multiples,  then  these  spher- 
ical surfaces  will  repre- 


FIG.  185. 


sent  the  lateral 

of  waves  which  are  diffused  on  every  side  of  the  direct  beam, 


MUSICAL    SOUNDS.  197 

B  D,  C  E.  But  the  sound  is  in  general  more  feeble  as  the  distance 
from  B  D,  or  C  E,  is  greater,  and  in  certain  points  is  destroyed 
by  interference.  This  spreading  of  sound  in  lateral  directions  is 
called  the  inflection  of  sound. 

What  is  true  of  all  sides  of  an  opening  is  of  course  true  when- 
ever sound  passes  by  the  side  of  an  obstacle.  Instead  of  being 
limited  by  lines  almost  straight  drawn  from  the  source,  as  light  is 
in  the  formation  of  a  shadow,  it  bends  round  the  edge,  and  is 
heard,  though  more  feebly,  behind  the  intervening  body.  It  has 
been  already  noticed  (Art.  288)  that  in  water  there  is  little  or  no 
inflection  of  sound. 


CHAPTEE   III. 

MUSICAL  SOUNDS  AND  MODES  OF  PRODUCING  THEM. 

300.  The  Vibrations  in  Musical  Sounds.— When  the  im- 
pulses of  a  sounding  body  upon  the  air  are  equidistant,  and  of  suf- 
ficient frequency,  they  produce  what  is  termed  a  musical  sound. 
In  most  cases  these  impulses  are  the  isochronous  vibrations  of  the 
body  itself,  but  not  necessarily  so ;  it  is  found  by  experiment  that 
blows  or  pulses,  of  any  species  whatever,  if  they  are  more  than 
about  15  or  20  per  second,  and  possess  the  property  of  isochronism, 
cause  a  musical  tone.     For  example,  the  snapping  of  a  stick  on 
the  teeth  of  a  metallic  wheel  would  seem  as  unlikely  as  anything 
to  produce  a  musical   sound;    but  when  the  wheel  is  in  rapid 
motion,  the  succession  causes  a  pure  musical  note.    Equidistant 
echoes  often  produce  a  musical  sound,  as  when  a  person  stamps  on 
the  floor  of  a  rectangular  room,  finished,  but  unfurnished  (Art. 
296).     So,  on  a  walk  by  the  side  of  a  long  baluster  fence,  a  sud- 
den sharp  sound,  like  the  blow  of  a  hammer  on  a  stone,  brings 
back  a  tone  more  or  less  prolonged,  resembling  the  chirp  of  a  bird. 
It  is  occasioned  by  successive  equidistant  echoes  from  the  balusters 
of  the  fence.    A  flight  of  steps  will  sometimes  produce  the  same 
effect,  the  tone  being  on  a  lower  key  than  that  from  the  fence, 
as  it  should  be. 

301.  The  Pitch  of  Musical  Sounds.— What  is  called  the 
pitch  of  a  musical  sound,  or  its  degree  of  acuteness,  is  owing  en- 
tirely to  its  rate  of  vibration.    Other  qualities  of  sounds  are  due 
to  other  and  often  unknown  circumstances ;  but  rapidity  of  vibra- 
tion is  the  only  condition  on  which  the  pitch  depends.     In  com- 


198  ACOUtfTICb. 

paring  one  musical  sound  with  another,  if  the  number  of  vibra- 
tions per  second  is  greater,  the  sound  is  more  acute,  and  is  said 
to  be  of  a  higher  pitch  ;  if  the  vibrations  are  fewer  per  second,  the 
sound  is  graver,  or  of  a  lower  pitch. 

302.  The  Monochord  or  Sonometer. — If  a  string  of 
uniform  size  and  texture  is  stretched  on  a  box  of  thin  wood,  by 
means  of  a  pulley  and  weight,  the  instrument  is  called  a  mono- 
chord,  and  is  useful  for  studying  the  laws  of  vibrations  in  musical 
sounds.  The  sound  emitted  by  the  vibrations  of  the  whole  length 
of  the  string  is  called  its  fundamental  sound. 

If  the  string  be  drawn  aside  from  its  straight  position,  and 
then  released,  one  component  of  the  force  of  tension  urges  every 
particle  back  towards  its  place  of  rest ;  but  the  string  passes  be- 
yond that  place,  on  account  of  the  momentum  acquired,  and  de- 
viates as  far  on  the  other  side  ;  from  which  position  it  returns,  for 
the  same  reason  as  before,  and  continues  thus  to  vibrate  till  ob- 
structions destroy  its  motion.  By  the  use  of  a  bow,  the  vibrations 
may  be  continued  as  long  as  the  experimenter  chooses. 

The  pitch  of  the  fundamental  sound  of  musical  strings  is  found 
by  experience  to  depend  on  three  circumstances  ;  the  length  of  the 
string — its  weight  or  quantity  of  matter — and  its  tension.  The 
tone  becomes  more  acute  as  we  increase  the  tension,  or  diminish 
either  the  length  or  the  weight.  The  operation  of  these  several 
circumstances  may  be  seen  in  a  common  violin.  The  pitch  of  any 
one  of  the  strings  is  raised  or  lowered  by  turning  the  screw  so  as 
to  increase  or  lessen  its  tension ;  or,  the  tension  remaining  the 
same,  higher  or  lower  notes  are  produced  by  the  same  string,  by 
applying  the  fingers  in  such  a  manner  as  to  shorten  or  lengthen 
the  string  which  is  vibrating  ;  or,  both  the  tension  and  the  length 
of  the  string  remaining  the  same,  the  pitch  is  altered  by  making 
the  string  larger  or  smaller,  and  thus  increasing  or  diminishing 
its  weight. 

A  string  is  said  to  make  a  single  vibration  in  passing  from  the 
extreme  limit  on  one  side  to  the  extreme  limit  on  the  other ;  a 
double  vibration  is  the  motion  across  and  back  again  to  the  origi- 
nal position.  Independently  of  calculation,  it  is  easy  to  see  that, 
with  a  given  weight  per  inch,  and  a  given  tension,  the  string  will 
vibrate  slower,  if  longer,  since  there  is  more  matter  to  be  moved, 
and  only  the  same  force  to  move  it ;  and  for  a  similar  reason,  the 
length  and  tension  being  given,  it  will  also  vibrate  slower,  if 
heavier.  On  the  other  hand,  if  length  and  weight  are  given,  it 
will  vibrate  faster,  if  the  tension  is  greater  ;  because  a  greater  force 
will  move  a  given  quantity  at  a  swifter  rate. 


199 

303.  Time  of  a  Single   Vibration.  —  The  mathematical 
formula  for  the  time  of  a  single  vibration  is 

T  - 

T- 

in  which  T  is  the  time,  in  seconds,  of  a  vibration  ;  I  =  length  of 
the  string  ;  w  =  the  weight  of  one  unit  of  the  string  ;  t  =  the 
tension,  and  g  =  the  force  of  gravity.  In  applying  the  formula 
I  and  g  must  be  in  the  same  unit,  either  both  in  feet  or  both  in 
inches,  and  also  w  and  t  in  the  same  unit,  both  in  ounces  or  both 
in  pounds. 

The  constant  factor,  g,  being  omitted,  the  variation  may  be 
expressed  thus  : 


T  oc    -;  that  is, 

Vt 

The  time  of  a  vibration  varies  as  the  length  of  the  string  multi- 
plied by  the  square  root  of  its  weight  per  inch,  and  divided  by  the 
square  root  of  its  tension. 

As  the  distance  of  the  string  from  its  quiescent  position  does 
not  form  an  element  of  the  algebraic  expression  for  the  time  of  a 
vibration,  it  follows  that  the  time  is  independent  of  the  amplitude. 
Hence,  as  in  the  pendulum,  the  vibrations  of  a  string,  fixed  at 
both  ends,  are  performed  in  equal  times,  whether  the  amplitude 
of  the  vibrations  be  greater  or  smaller.  It  is  on  this  account  that 
the  pitch  of  a  string  does  not  alter,  when  left  to  vibrate  till  it 
stops.  The  excursions  from  side  to  side  grow  less,  and  therefore 
the  sound  more  feeble,  till  it  ceases  ;  but  the  rate  of  vibration,  and 
therefore  the  pitch,  remains  the  same  to  the  last.  This  property 
of  isochronism,  independent  of  extent  of  excursion,  is  common  to 
sounding  bodies  generally,  and  is  owing  to  what  may  be  called  the 
law  of  elasticity,  that  the  restoring  force,  acting  on  any  particle, 
varies  directly  as  its  distance  from  the  place  of  rest.  For  example, 
each  particle  of  the  string,  if  removed  twice  as  far  from  its  place 
of  rest,  is  urged  back  by  a  force  twice  as  great,  and  therefore  returns 
in  the  same  time. 

304.  The  Number  of  Vibrations  in  a  Given  Time.—  The 
greater  is  the  length  of  one  vibration,  the  less  will  be  the  number 
of  vibrations  in  a  given  time  ;  that  is,  if  N  represents  the  number, 

JVoc  •=-;  but  as  T  oc        _  ,  .-.  JV^oc  -  —  .     If  t  and  w  are  con- 
1  V  t  I  Vw 

stant,  N  oc  -  ;  if  I  and  t  are  constant,  N  oc       _  ;  and  if  I  and  w 
1  v  w 

are  constant,  N  oc   \/7;  that  is, 

1.  The  number  of  vibrations  varies  inversely  as  the  length. 


200  ACOUSTICS. 

2.  TJie  number  of  vibrations  varies  inversely  as  the  square  root 
of  the  weight  of  the  string. 

3.  The  number  of  vibrations  varies  as  the  square  root  of  the 
tension. 

Thus,  the  number  of  vibrations  in  a  second  may  be  doubled, 
either  by  halving  the  length  of  the  string  or  by  making  its  weight 
one-fourth  as  great,  or,  finally,  by  making  its  tension  four  times 
as  great. 

305.  Vibrations  of  a  String  in  Parts.— The  monochord 
may  be  made  to  vibrate  in  parts,  the  points  of  division  remaining 
at  rest;  and  this  mode  of  vibration  may  even  coexist  with  the  one 
already  described.  Of  course  the  sound  produced  by  the  parts 
will  be  on  a  higher  pitch,  since  they  are  shorter,  while  the  tension 
and  the  weight  per  inch  remain  unaltered.  It  is  a  noticeable  fact 
that  the  parts  are  always  such  as  will  exactly  measure  the  whole 
without  a  remainder.  Hence  the  vibrating  parts  are  either  halves, 
thirds,  fourths,  or  other  aliquot  portions.  The  sounds  produced 
by  any  of  these  modes  of  vibration  are  called  harmonics,  for  a 
reason  which  will  appear  hereafter.  Suppose  a  string  (Fig.  186)  to 

FIG.  186. 


be  stretched  between  A  and  B,  and  that  it  is  thrown  into  vibra- 
tion in  three  parts.  Then  while  A  D  makes  its  excursion  on  one 
side,  D  C  will  move  in  the  opposite  direction,  and  C  B  the  same 
as  A  D\  and  when  one  is  reversed,  the  others  are  also,  as  shown 
by  the  dotted  line.  In  this  way  D  and  C  are  kept  at  rest,  being 
urged  toward  one  side  by  one  portion  of  string,  and  toward  the 
opposite  by  the  next  portion.  But  the  string  may  at  the  same 
time  vibrate  as  a  whole  ;  in  which  case  D  and  C  will  have  motion 
to  each  side  of  their  former  places  of  rest,  while  relatively  to  them 
the  three  portions  will  continue  their  movements  as  before.  The 
points  C  and  D  are  called  nodes ;  the  parts  A  D,  D  C,  and  C  B, 
are  called  ventral  segments.  By  a  little  change  in  the  quickness 
of  the  stroke,  the  bow  may  be  made  to  bring  from  the  monochord 
a  great  number  of  harmonic  notes,  each  being  due  to  the  vibra- 
tions of  certain  aliquot  parts  of  the  string.  By  confining  a  partic- 
ular point,  however,  at  the  distance  of  -J-,  J,  or  other  simple  frac- 
tion of  the  whole  from  the  end,  the  particular  harmonic  belonging 
to  that  mode  of  division  may  be  sounded  clear,  and  unmingled 
with  the  others. 

306.  Longitudinal  Vibrations  of  Strings.— A  string  may 


VIBRATIONS    OF    AIR    COLUMNS.  201 

vibrate  in  the  direction  of  its  length,  in  consequence  of  the  elastic 
force  of  its  molecules.  Such^  vibrations  can  be  produced  by  rub- 
bing the  stretched  string  quickly  with  resined  leather  in  the 
direction  of  its  length. 

The  fundamental  note  thus  produced  is  of  much  higher  pitch 
than  that  of  the  same  string  caused  to  vibrate  transversely,  owing 
to  the  great  molecular  elasticity  as  compared  with  the  elasticitv 
due  to  tension  of  the  string  between  its  supports. 

By  experimenting  as  in  the  case  of  transverse  vibrations  it  may 
be  shown  that  the  number  of  vibrations  in  a  given  time  is  in- 
versely proportional  to  the  length  of  the  string. 

By  altering  the  tension  within  the  limits  of  elasticity  of  the 
substance  no  change  of  pitch  is  produced,  longitudinal  vibrations 
differing  in  this  respect  from  transverse. 

Changing  the  thickness  or  weight,  the  material  being  the 
same,  does  not  alter  the  pitch,  another  difference  to  be  noted  be- 
tween transverse  and  longitudinal  vibrations. 

Two  wires  of  different  material  but  of  the  same  length  will 
give  different  notes.  The  time  required  for  a  pulse  to  run  from 
one  end  to  the  other  and  back  again  is  the  same  as  the  time  of  a 
complete  vibration  of  any  one  molecule  ;  hence  in  a  long  wire  the 
vibrations  must  be  slower  than  in  a  short  one,  since  it  will  take 
the  pulse  a  longer  time  to  travel  the  length  of  the  long  wire  than 
to  move  the  length  of  a  short  one. 

Now  if  two  wires  of  the  same  length  but  of  different  materials 
be  used,  that  which  transmits  the  sound  pulse  with  the  greatest 
velocity  will  give  the  highest  note,  since  the  number  of  vibrations 
per  second  due  to  this  greater  velocity  of  transmission  will  be 
greater.  If  two  wires  of  different  materials  be  so  adjusted  as  to 
length  as  to  give  the  same  note,  the  ratio  of  their  lengths  is  the 
ratio  of  the  velocities  of  transmission  of  sound  in  the  two  sub- 
stances. 

307.  Vibrations  of  a  Column  of  Air.— When  a  musical 
sound  is  produced  by  a  pipe  of  any  kind,  it  is  the  column  of  in- 
closed air  which  must  be  regarded  as  the  sounding  body.  A  con- 
densed wave  is  caused,  by  some  mode  of  excitation,  to  travel  back 
and  forth  in  the  pipe,  followed  by  a  rarefied  portion  ;  and  these 
waves  affect  the  surrounding  air  much  in  the  same  way  as  do  the 
alternate  excursions  of  a  string.  That  it  is  the  air,  and  not  the 
pipe  itself,  which  is  the  source  of  sound,  is  proved  by  using  pipes 
of  various  materials — the  most  elastic  and  the  most  inelastic — as 
glass,  wood,  paper,  and  lead ;  if  they  are  of  the  same  form  and 
size,  the  tone  in  each  case  has  the  same  pitch. 

In  order  to  examine  the  manner  in  which  the  air-columns  in 


202 


ACOUSTICS. 


FIG.  187. 


pipes  perform  their  vibrations,  it  is  convenient  to  consider  them 
in  three  classes  : 

1st.  Pipes  which  are  closed  at  one  end  and  open  at  the  other. 

2d.  Those  which  are  closed  at  both  ends. 

3d.  Those  which  are  open  at  both  ends. 

308.  One  End  of  the  Pipe  Closed.— Suppose  the  column 
of  air  in  the  pipe  B  A  (Fig.  187)  to  be  of  such  length  as  to  re- 
spond to  the  motions  of  the  fork  placed  above  it, 
in  which  case  its  length  would  be  almost  exactly 
one-fourth  the  length  of  the  sound  wave  pro- 
duced by  the  fork  ;  then  while  the  prong  moves 
from  a  to  b  the  condensed  pulse  travels  from 
B  to  A  and  back  again  to  B,  having  been  re- 
flected at  A,  reaching  B  just  as  the  prong  begins 
its  excursion  back  again  to  a,  producing  a  rare- 
faction which  in  like  manner  travels  to  A  and 
back  again  to  B  by  the  time  the  prong  has 
reached  a,  ready  to  begin  the  next  vibration.  These  successive 
condensations  and  rarefactions  passing  down  and  up  the  air 
column  produce  the  sound  or  note  peculiar  to  the  pipe  in  use. 


309.  Both    Ends 
pipes,  like  that  used  in 

FIG.  188. 


1 

\ 

C 

B    B' 

? 

of  the  Pipe  Closed. — If  two  equal 
the  last  paragraph,  be  placed  with  their 
open  ends  near  together  as 
in  Fig.  188,  and  a  vibrating 
prong  or  reed  communicates 
motion  to  the  air  columns 
as  before,  while  the  reed 
moves  from  a  to  ft  a  con- 
densed pulse  will  pass  from 
B'  to  C'  and  back  again, 
while  at  the  same  time  a  rarefied  pulse  will  travel  from  B  to 
C  and  back  again ;  while  the  reed  moves  from  b  to  a  the  con- 
densation passes  down  B  C  and  back,  while  the  rarefaction  moves 
down  B'  C'  and  back  again.  Each  separate  pipe  in  this  case 
gives  the  same  note  as  though  it  alone  were  used.  Now  if  we  join 
the  pipes  at  B  B'  leaving  only  a  small  orifice  through  which  to 
communicate  motion  to  the  air  column,  the  effect  is  in  no  way 
changed,  and  the  note  will  still  be  the  same.  Thus  in  Fig.  189, 
while  the  condensed  pulse  moves  from  B  to  C,  the  point  of  rare- 
faction runs  from  A  to  (7,  where  they  pass  each  other ,  hence,  at 
the  middle  of  the  pipe  there  is  no  change  of  density,  since  every 
degree  of  condensation  is  at  that  point  met  by  an  equal  degree  of 
rarefaction  of  the  other  half  of  the  general  wave.  At  the  ex- 


VIBRATIONS    OF    AIR    IN     PIPES.  203 

tremities,  A  and  J9,  there  is  alternately  a  maximum  of  conden- 
sation and  of  rarefaction,  each  being  reflected  and  returning,  to 
meet  again  at  C.     Fig.  189  shows  the  air  in  a  state  of  condensa- 
tion at  A,  and  of  rarefac- 
tion at  B.     At  all  points 
between  the    centre  and 
the  ends  there  is  alternate 
condensation  and  rarefac- 
tion, but  in  a  less  degree 
according  to  the  distance  from  the  ends. 

On  the  other  hand,  the  excursions  of  the  particles  are  greatest 
at  C,  and  nothing  at  A  and  B,  where  all  motion  is  prevented  by 
the  fixed  stoppers  by  which  the  pipe  is  closed.  Between  the  ends 
and  the  centre,  the  amplitude  of  vibration  is  greater,  as  the  dis- 
tance from  the  centre  is  less. 

The  pitch  of  such  a  pipe  will  be  lower,  as  the  pipe  is  longer, 
because  the  waves  have  a  greater  distance  to  travel  between  the 
successive  reflections,  and  hence  there  will  be  a  smaller  number 
per  second.  So  also,  lowering  the  temperature  lowers  the  pitch, 
since  the  wave  then  travels  more  slowly,  and  suffers  fewer  reflec- 
tions in  a  second. 

310.  Both  Ends  of  the  Pipe  Open.— When  both  ends  of 
a  pipe  are  open,  it  may  still  produce  a  musical  tone,  by  having  a 
node  in  the  centre  of  it,  thus  forming  two  pipes  like  the  one  first 
described.    When  the  vibration  is  established  in  such  a  pipe,  the 
pulses  from  the  ends  move  simultaneously  toward  C  (Fig.  190), 
and  again  from  it  after  re- 
flection.   Thus  C  is  a  fixed       t **"*•  m 

point,  where  the  greatest 

condensation  and  rarefac- 
tion occur  alternately,  like 
A  in  Fig.  187.  It  there- 
fore has  the  same  pitch  as  A  C  alone,  stopped  at  C  and  open  at  A. 
If  a  solid  partition  be  inserted  at  C,  it  causes  no  change  of  pitch. 
Such  a  pipe  can  produce  no  sound,  except  by  the  formation  of 
at  least  one  node. 

311.  The  First    Kind    of   Pipe    is    the    Elementary 
Form. — In  comparing  with  each  other  the  three  kinds  of  pipe 
which  have  been  described,  it  is  observable  that  the  second  kind 
(stopped  at  both  ends),  and  the  third  kind  (open  at  both  ends), 
are  both  double  pipes  of  the  first    kind  (open  at  one  end,  and 
stopped  at  the  other).      For,  if  two  pipes  of  the  first  kind  be 
placed  with  their  open  ends  together,  as  we  have  seen,  they  form 
one  of  the  second  kind,  and  there  is  no  change  of  pitch.    Again,  if 


C 


204  ACOUSTICS. 

the  two  be  placed  with  the  closed  ends  in  contact,  they  form  a  pipe 
of  the  third  class ;  since  the  partition  may  remain  or  be  removed, 
without  affecting  the  mode  of  vibration.  Hence,  a  pipe  open  at 
both  ends,  and  one  of  the  same  length  closed  at  both  ends,  each 
yields  the  same  fundamental  note  as  a  pipe  of  half  their  length, 
open  only  at  one  end. 

312.  Vibrations  of  a  Column  of  Air  in  Parts.— The 
same  is  true  of  a  column  of  air  as  of  a  string,,  that  it  may  vibrate 
in  parts  ;  and  also  that  two  or  more  modes  of  vibration  may  co- 
exist in  the  same  column. 

The  second  and  third  kinds  of  pipe  can  divide  so  that  the  whole 
and  the  vibrating  segments  have  the  ratios  of  1  :  J  :  -J  :  &c. ;  these 
ratios  in  the  closed  pipe  are  shown  in  Figs.  188,  191,  and  192 ;  and 
in  the  open  pipe  in  Figs.  190  and  193.  In  Fig.  191  the  pipe  is 

divided  into  two  equal 
parts,  in  each  of  which 
the  vibrations  take  place 
in  the  same  manner  as 
in  the  whole,  Fig.  188. 
Condensations  run  si- 
multaneously from  A  and  B  to  the  middle  point  (7,  and  thence 
back  to  A  and  B.  When  C  is  condensed,  A  and  B  are  rarefied  ; 
and  when  A  and  B  are  condensed,  (7  is  rarefied.  Those  three 
points  have  no  amplitude,  but  the  greatest  changes  in  density. 
But  the  points  midway  between  have  the  greatest  amplitude,  and 
no  change  of  density.  As  the  waves  run  over  the  parts  in  half 
the  time  that  they  would  over  the  whole,  the  pitch  is  raised  ac- 
cordingly. In  this  mode  of  vibrating,  the  opening  where  the 
pIG  igg  vibrations  are  excited 

cannot  be   at    C,   where 
the  node  is  formed. 

In  Fig.  192  are  shown 


— I 


A.  jj         c        JS  JS    three  vibrating  segments. 

B  and  D  are  condensed  at  one  moment,  A  and  E  at  another. 

In  the  third  kind,  as  already  stated  (Art.  310),  there  must  be 
at  least  one  node.     When  there  are  two,  it  is  apparent  by  Fig.  193 

that  they  must  be   one- 
fourth  of  the  length  from 
=       each  end,  in  order  that  the 
three  parts  may  vibrate 
_      in  unison  ;  for  the  middle 

A  C  JE  _»  JS 

part  is  a  complete  seg- 
ment, like  the  pipe  A  B  (Fig.  189),  while  the  ends  are  half  seg- 
ments, like  the  pipe  A  B  (Fig.  187).  If  there  were  three  nodes, 


POSITION    OF    NODES. 


205 


FIG.  194. 


FIG.  195. 


there  would  be  two  complete  segments  between  them,  and  two 
half  segments  at  the  ends.  It  is  evident  that  the  lengths  of  the 
half  segments,  being  £,  J-,  -J-,  &c.,  are  as  1,  £,  J,  &c.,  of  the  whole 
pipe ;  therefore  the  rates  of  vibration  (being  inversely  as  the 
lengths)  are  as  the  numbers  1,  2,  3,  &c. 

The  length  of  the  elementary  form  of  pipe  is  one-fourth  the 
length  of  the  fundamental  sound  wave  which  it  produces.  In 
this  form  of  pipe  nodes  may  divide  the  pipe  into  segments  having 
the  ratios  1  :  £  :  -J-,  &c.  The  simplest  division  is  by  one  node,  a 
third  of  the  length  from  the  open  end,  as  in  Fig.  194.  Then  C  D, 
a  half  segment,  and  A  D,  a  complete  seg- 
ment, have  the  same  rate  of  vibration.  If 
there  were  two  nodes,  one  must  be  a  fifth 
f rom  the  open  end,  while  the  other  divides 
the  remainder  into  two  complete  segments.  J 
Therefore,  in  the  several  modes  of  vibra- 
tion of  the  first  kind  of  pipe,  the  half  segments,  being  1,  -J-,  £,  &c., 
of  the  whole  length,  the  rates  of  vibration  in  them  are  as  the  odd 
numbers  1,  3,  5,  &c. 

313.  Position  of  Nodes  Determined  Experimentally.— 

The  position  of  nodes  in  pipes  may  be  readily  shown 
by  introducing  a  ring  R  (Fig.  195)  over  which  is 
stretched  a  thin  membrane.  If  this  ring,  suspended 
from  a  cord,  be  lowered  into  the  open  upper  end  of  a 
sounding  pipe  of  glass,  or  one  which  has  one  trans- 
parent face,  it  will  make  a  rattling  or  fluttering 
noise,  or  will  show  that  it  is  vibrating  by  the  motion 
of  grains  of  sand  sprinkled  upon  it ;  this  vibration 
decreases  in  intensity  as  the  disc  is  gradually  lowered, 
until,  when  the  disc  reaches  the  place  of  a  node,  the 
vibration  ceases  and  the  sand  remains  at  rest.  Thus 
the  place  of  each  of  any  number  of  nodes  may  be 
determined  experimentally.  Fine  silica  powder  in 
a  sounding  pipe  held  horizontally  will  also  mark 
the  segments  and  nodes  very  beautifully. 

314.  Modes    of   Exciting    Vibrations    in 
Pipes. — There  are  two  methods  of  making  the  air- 
column  in  a  pipe  to  vibrate :  one  by  a  stream  of  air 
blown  across  an  orifice  in  the  pipe,  the  other  by  an 
elastic  plate  called  a  reed.     A  familiar  example  of 
the  first  is  the  flute.     A  stream  of  air  from  the  lips 
is  directed  across  the  embouchure,  so  as  just  to  strike 
the  opposite  edge;  this  causes  a  .wave  to  move 
through  the  tube.     The  stream  of  air,  like  a  spring, 


A 


206 


ACOUSTICS. 


FIG.  196. 


IG.  198. 


vibrates  so  as  to  keep  time  with  the  movement  of  the  wave  to  and 
fro,  while  at  each  pulse  it  renews  that  movement,  and  makes  the 
sound  continuous.  For  higher  notes,  the  stream  must  be  blown 
more  swiftly,  that  by  its  greater  elastic  force,  it  may  be  able  to 
conform  to  the  more  rapid  vibration  of  the  column. 
A  large  proportion  of  the  pipes  of  an  organ  are  made 
to  produce  musical  tones  essentially  in  the  same  way 
as  the  flute,  and  are  called  mouth-pipes. 

Fig.  196  shows  the  construction  of  the  mouth-pipe 
of  an  organ  ;  o  b  is  the  mouth  ;  and  as  the  stream  of  air 
issues  from  the  channel  i,  it  starts  a  wave  in  the  pipe, 
and  then  the  stream  itself  vibrates  laterally  past  the 
lip  #,  keeping  time  with  the  successive  returns  of  the 
wave  in  the  pipe.  The  pipe  is  attached  to  the  wind- 
chest  by  the  foot  P. 

The  clarinet  is  an  example  of  vibrations  in  an  air- 
column  by  a  reed.     In  that  instrument  the  reed   is 
often  made  of  wood;  when  the  air  is  blown  past  its 
edge  into  the  tube,  the  reed  is  thrown  into  vibration, 
and  by  it  the  column  of  air.    The  strength  of  elasticity  in  the  reed 

should  be  such  that  its  vibrations  will 
keep  j.-me  W|tj1  ^e  excursjons  Of  the 

wave  in  the  column.  What  are  called 
the  reed  pipes  of  the  organ  are  con- 
structed on  the  same  principle,  but 
the  reeds  are  metallic.  An  example 
is  seen  in  Fig.  197,  which  represents 
a  model  of  the  reed  pipe,  made  to 
show  the  vibrations  through  the  glass 
walls  at  E.  A  chimney,  IT,  is  usually 
attached,  sometimes  of  a  form  (as 
in  the  figure)  to  increase  the  loud- 
ness  of  the  sound,  and  sometimes  of 
a  different  form,  for  softening  it. 
The  air  in  an  open  tube  may  also  be 
thrown  into  vibration  by  a  burning 
jet  of  gas,  as  in  Fig.  198.  The  note 
depends  upon  the  size  of  the  flame 
and  the  length  of  the  tube.  By  vary- 
ing the  position  of  the  jet  in  a  long 
tube  a  series  of  nodes  in  the  ratio  of 
1:2:3:4,  &c.,  can  be  obtained. 

If  a  gauze  diaphragm  be  inserted 
in  a  tube  open  at  both  ends,  about  two 
inches  in  diameter  and  two  feet  long, 


PIG.  197. 


WIRES.  207 

at  a  point  three  or  four  inches  from  the  end,  and  this  diaphragm 
be  heated  red  hot  by  a  Bunsen  burner,  upon  removing  the  burner 
and  depressing  the  tube  so  as  to  cause  a  current  of  air  to.  pass 
through  it,  a  very  loud  note  will  be  produced. 

315.  Vibrations   of  Rods   and    Laminae. — A    plate    of 
metal  called  a  reed  is  much  used  for  musical  purposes  in  connec- 
tion with  a  column  of  air,  as  already  stated.     In  parlor  organs 
the  sound  is  produced  by  the  action  of  vibrating  reeds  upon  air 
currents,  just  as  in  the  case  of  the  musical  toy  called  harmonieon. 
Except  in  such  connection,  the  sounds  of  wires  and  laminae  are 
generally  too  feeble  to  be  employed  in  music.     But  their  vibra- 
tions have  been  much  studied,  on  account  of  the  interesting  phe- 
nomena attending  them. 

Such  vibrations  afford  a  convenient  mode  of  determining  the 
velocity  of  sound  in  solids.  A  rod,  held  firmly  by  a  clamp  in  the 
middle,  and  rubbed  about  half  way  between  the  middle  and  the 
end  by  a  leather  pad  well  resined,  will  give  a  note  due  to  longi- 
tudinal vibrations  of  the  rod.  While  the  rod  gives  out  its  funda- 
mental note  the  ends  vibrate  freely,  being  neither  compressed  nor 
extended  :  but  at  the  centre,  held  by  the  clamp,  there  is  no  vibra- 
tion, but  a  maximum  effect  of  alternate  compression  when  the  two 
pulses  meet,  and  extension  when  they  again  travel  toward  the 
ends.  The  middle  of  the  rod  is  a  node.  The  time  of  a  complete 
vibration  is  the  time  that  a  pulse  would  require  to  travel  twice 
the  length  of  the  rod.  If  the  note  given  is  due  to  512  vibrations 
per  second,  and  the  length  of  the  rod  be  x  feet,  then  a  length  of 
%x  feet  is  passed  over  512  times  in  a  second,  and  the  velocity  in 
the  substance  of  the  rod  is  512  x  2x  feet  per  second. 

316.  Wires. — If  one  end  of  a  steel  wire  is  fastened  in  a  vise 
and  vibrated,  while  a  thin  blade  of  sunlight  falls  across  it,  the 
path  of  the  illuminated  point  may  be  traced.     It  is  not  ordinarily 
a  circular  arc  about  the  fixed  point  as  a  centre,  but  some  irregular 
figure  ;  and  frequently  the  point  describes  two  systems  of  ellipses, 
the  vibrations  passing  alternately  from  one  system  to  the  other 
several  times  before  running  down.     If  the  structure  of  the  wire 
were  the  same  in  every  part  across  its  section,  and  if  the  fastening 
pressed  equally  on  every  point  around  it,  the  orbit  of  each  particle 
would  be  a  series  of  ellipses,  whose  major  axes  are  on  the  same 
line.     If,  moreover,  there  was  no  obstruction  to  the  motion,  and 
the  law  of  elasticity  could  obtain  perfectly,  it  would  vibrate  in  the 
same  elliptic  orbit  forever,   the  force  toward  the  centre  beinjL 
directly  as  the  distance.    It  is  easy  to  cause  the  wire,  in  the  ex 
periment  just  described,  to  vibrate  also  in  parts ;  in  which  cas* 
each  atom,  while  describing  the  elliptic  orbit,  will  perform  seven 


208  ACOUSTICS. 

smaller  circuits,  which  appear  as  waves  on  the  circumference  of  the 
larger  figure. 

317.  Chladni's  Plates. — If  a  square  plate  of  glass  or  elastic 
metal,  of  uniform  thickness  and  density,  be  fastened  by  its  centre 
in  a  horizontal  position,  and  a  bow  be  drawn  on  its  edge,  it  will 
emit  a  pure  musical  tone  ;  and  by  varying  the  action  of  the  bow, 
and  touching  different  points  of  the  edge  with  the  finger,  a  variety 
of  sounds  may  be  obtained  from  it.    The  plate  necessarily  vibrates 
in  parts ;  and  the  lowest  pitch  is  produced  when  there  are  two 
nodal  lines  parallel  to  the  sides,  and  crossing  at  the  centre,  thus 
dividing  the  plate  into  four  square  ventral  segments.     The  posi- 
tion of  the  nodal  lines,  and  the  forms  of  the  segments,  are  beauti- 
fully exhibited  by  sprinkling  writing-sand  on  the  plate.    The  par- 
ticles will  dance  about  rapidly  till  they  find  the  lines  of  rest,  where 
they  will  presently  be  collected.     For  every  new  tone  the  sand  will 
show  a  new  arrangement  of  nodal  lines ;   and  as  two  or  more 
modes  of  vibration  may  coexist  in  plates,  as  well  as  in  strings  and 
columns  of  air,  the  resultant  nodes  will  also  be  rendered  visible. 
Again,  by  fastening  the  plate  at  a  different  point,  still  other  ar- 
rangements will  take  place,  each  distinguishable  by  the  position 
of  its  nodal  lines  and  the  pitch  of  its  musical  note.     The  form  of 
the  plate  itself  may  also  be  varied,  and  each  form  will  be  charac- 
terized by  its  own  peculiar  systems.     Chladni,  who  first  performed 
these  interesting  experiments,  delineated  and  published  the  forms 
of  ninety  different  systems  of  vibration  in  the  square  plate  alone. 

If  a  fine  light  powder,  as  lycopodium  (the  pollen  of  a  species 
of  fern),  be  scattered  on  the  plate,  it  is  affected  in  a  very  different 
manner  from  heavy  sand.  It  will  gather  into  rounded  heaps  on 
those  portions  of  the  segments  which  have  the  greatest  amplitude 
of  vibration  ;  the  particles  which  compose  the  heaps  performing  a 
continual  circulation,  down  the  sides  of  the  heaps,  along  the  plate 
to  the  centre,  and  up  the  axis.  If  the  vibration  is  violent,  the 
heaps  will  be  thrown  up  from  the  plate  in  little  clouds  over  the 
portions  of  greatest  motion.  The  cause  of  this  singular  effect  was 
ascertained  by  Faraday,  who  found  that  in  an  exhausted  receiver 
the  phenomenon  ceased.  It  is  due  to  a  circulation  of  the  air, 
which  lies  in  contact  with  a  vibrating  plate.  The  air  next  to 
those  parts  which  have  the  greatest  amplitude  is  at  each  vibration 
thrown  upward  more  powerfully  than  elsewhere,  and  surround- 
ing particles  press  into  its  place,  and  thus  a  circulation  is  estab- 
lished ;  and  a  fine  light  powder  is  more  controlled  by  these  at- 
mospheric movements  than  by  the  direct  action  of  the  plate. 

318.  Bells. — If  a  thin  plate  of  metal  takes  the  form  of  a  cylin- 
der or  bell,  its  fundamental  note  is  produced  when  each  ring  of  the 


THE    VOICE.  209 

material  changes  from  a  circle  to  an  ellipse,  and  then  into  a  second 
ellipse,  whose  axis  is  at  right  angles  to  that  of  the  former,  as  seen  in 
Fig.  199.  It  thus  has  four  ventral  seg- 
ments and  four  nodal  lines,  the  latter  lying  IG* 
in  the  plane  of  the  axis  of  the  bell  or  cylin- 
der. If  the  rings  which  compose  the  bell 
were  all  detached  from  one  another,  they 
would  have  different  rates  of  vibration  ac- 
cording to  their  diameter,  and  hence  would 
produce  tones  of  various  pitch  ;  but,  being 
bound  together  by  cohesion,  they  are  com- 
pelled to  keep  the  same  time,  and  hence 
give  but  one  fundamental  tone.  But  a 
bell,  especially  if  quite  thin,  may  be  made  to  emit  a  series  of  har- 
monic sounds  by  dividing  up  into  a  greater  number  of  segments. 
It  is  obvious  that  the  number  of  nodes  must  always  be  even,  be- 
cause two  successive  segments  must  move  in  opposite  directions 
in  one  and  the  same  instant ;  otherwise  the  point  between  them 
could  not  be  kept  at  rest,  and  therefore  would  not  be  a  node.  Be- 
sides the  principal  tone  of  a  church-bell,  one  or  two  subordinate 
sounds  on  a  different  pitch  may  usually  be  detected.  A  glass  bell, 
suitably  mounted  for  the  lecture-room,  will  yield  ten  or  twelve 
harmonics,  by  means  of  a  bow  drawn  on  its  edge. 

319.  The  Voice. — The  vocal  organ  is  complex,  consisting 
of  a  cavity  called  the  larynx,  and  a  pair  of  membranous  folds  like 
valves,  having  a  narrow  opening  between  them;  this  opening, 
called  the  glottis,  admits  the  air  to  the  larynx  from  the  wind-pipe 
below.  The  edges  of  these  valves  are  thickened  into  a  sort  of 
cord,  and  for  this  reason  the  apparatus  is  called  the  vocal  cords.  In 
the  act  of  breathing,  the  folds  of  the  glottis  lie  relaxed  and  sepa- 
rate from  each  other,  and  the  air  passes  freely  between  them,  with- 
out producing  vibration.  But  in  the  effort  to  form  a  vocal  sound, 
they  approach  each  other,  and  become  tense,  so  that  the  current 
of  air  throws  them  into  vibration.  These  vibrations  are  enforced 
by  the  consequent  vibrations  in  the  air  of  the  larynx  above  ;  and 
thus  a  fullness  of  sound  is  produced,  as  in  many 
musical  instruments,  in  which  a  reed,  and  the  air 
of  a  cavity,  perform  synchronous  vibrations,  and 
emit  a  much  louder  sound  than  either  could  do 
alone.  If  two  pieces  of  thin  india-rubber  be 
'  stretched  across  the  end  of  a  tube,  with  their 
edges  parallel,  and  separated  by  a  narrow  space, 
as  represented  in  Fig.  200,  the  arrangement  will 
give  an  idea  of  the  larynx  and  glottis  of  the  vocal 
14 


210  ACOUSTICS. 

organ.  If  air  be  forced  through,  a  sound  is  produced,  whose 
pitch  depends  on  the  size  of  the  tube  and  the  tension  of  the  valves. 
The  natural  key  of  a  person's  voice  depends  on  the  length  and 
weight  of  the  vocal  cords,  and  the  size  of  the  larynx.  The  yield- 
ing nature  of  all  the  parts,  and  the  ability,  by  muscular  action,  to 
change  the  form  and  size  of  the  cavity  and  the  tension  of  the 
valves,  give  great  variety  to  the  pitch,  and  the  power  of  adjusting 
it  with  precision  to  every  shade  of  sound  within  certain  limits. 
No  instrument  of  human  contrivance  can  be  brought  into  com- 
parison with  the  organ  of  voice.  After  the  voice  is  formed  by  its 
appropriate  organ,  it  undergoes  various  modifications,  by  means 
of  the  palate,  the  tongue,  the  teeth,  the  lips,  and  the  nose,  before 
it  is  uttered  in  the  form  of  articulate  speech. 

320.  The  Organ  of  Hearing. — The  principal  parts  of  the 
ear  are  the  following : 

1.  The  outer  ear,  E  a  (Fig.  201),  terminating  at  the  mem- 
brane of  the  tympanum,  m. 

FIG.  201. 


2.  The  tympanum,  a  cavity  separated  from  the  outer  ear  by  a 
membrane,  m,  and  containing  a  series  of  four  very  small  bones 
(ossicles),  b,  c,  o  and  s,  severally  called,  on  account  of  their  form, 
the  hammer,  the  anvil,  the  ball,  and  the  stirrup.    The  figure 
represents  the  walls  of  the  tympanum  as  mostly  removed,  in  order 
to  show  the  internal  parts.     This  cavity  is  connected  with  the 
back  part  of  the  mouth  by  the  Eustachian  tube,  d. 

3.  The  labyrinth,  consisting  of  the  vestibule,  v,  the  semicircular 
canals,  f,  and  the  cochlea,  g.     The  latter  is  a  spiral  tube,  winding 
two  and  a  half  times  round.     The  parts  of  the  labyrinth  are  exca- 
vated in  the  hardest  bone  of  the  body.     The  figure  shows  only  its 
exterior.    There  are  two  orifices  through  the  bone  which  sepa- 


MUSICAL    SCALES.  211 

rates  the  labyrinth  from  the  tympanum,  the  round  orifice,  e,  pass- 
ing into  the  cochlea,  and  the  oval  orifice,  s,  leading  to  the  vesti- 
bule. These  orifices  are  both  closed  by  a  thin  membrane.  The 
ossicles  of  the  tympanum  form  a  chain  which  connects  the  centre 
of  the  membrane,  m,  with  that  which  closes  the  oval  orifice.  The 
labyrinth  is  filled  with  a  liquid,  in  various  parts  of  which  float 
the  fibres  of  the  auditory  nerve. 

By  the  form  of  the  outer  ear,  the  waves  are  concentrated  upon 
the  membrane  of  the  tympanum,  thence  conveyed  through  the 
chain  of  bones  to  the  membrane  of  the  labyrinth,  and  by  that  to 
the  liquid  within  it,  and  thus  to  the  auditory  nerve,  whose  fibres 
lie  in  the  liquid. 


CHAPTER    IV. 

MUSICAL  SCALES— THE  RELATIONS  OF  MUSICAL  SOUNDS. 

321.  Numerical  Relations  of  the  Notes. — To  obtain  the 
series  of  notes  which  compose  the  common  scale  of  music,  it  is 
convenient  to  use  the  monochord.  Calling  the  sound,  which  is 
given  by  the  whole  length  of  the  string,  the  fundamental,  or  key 
note,  of  the  scale,  we  measure  off  the  following  fractions  of  the 
whole  for  the  successive  notes,  namely :  f ,  £ ,  f ,  f ,  f ,  ^,  -J.  If  the 
whole,  and  these  fractions,  are  made  to  vibrate  in  order,  the  ear 
will  recognize  the  sounds  as  forming  the  series  called  the  gamut, 
or  diatonic  scale. 

Now  as  the  number  of  vibrations  varies  inversely  as  the  length 
of  the  string,  the  number  of  vibrations  of  the»e  notes  respectively, 
expressed  in  fractions  of  the  number  of  vibrations  of  the  whole 
string,  which  we  will  call  1,  will  be  1,  f,  f,  f,  f,  f,  ty,  2. 

If  the  whole  string  vibrates  128  times  per  second,  f  of  the 
string  would  give  £  of  128  vibrations,  or  144  vibrations;  f  of  the 
string  would  give  £  of  128  vibrations  per  second,  or  160  vibrations. 
To  express  the  relative  number  of  vibrations  in  the  series  above, 
reduce  the  fractions  to  a  common  denominator  and  compare  their 
numerators,  and  we  have 

24,  27,  30,  32,  36,  40,  45,  48. 

Sounds  whose  vibrations  per  second  bear  to  each  other  the 
ratios  of  the  series  above  are  not  arbitrarily  chosen  to  form  the 
scale,  but  they  are  demanded  by  the  ear.  The  notes  correspond- 
ing to  the  series  are  named  according  to  tneir  place  in  the  series  ; 
thus  a  note  whose  vibrations  are  f£  or  the  vibrations  of  the 
fundamental,  is  called  the  third,  one  whose  vibrations  are  |f  i? 


212  ACOUSTICS. 

the  fifth,  and  that  whose  vibrations  are  f f,  or  twice  as  many  as 
those  of  the  fundamental,  is  the  eighth  or  octave. 

322.  Relations  of  the  Intervals.  —  An  interval  is  the 
relative  pitch  of  two  sounds,  and  its  numerical  value  is  expressed 
by  a  fraction  whose  numerator  is  the  number  of  vibrations  per 
second  of  the  higher  sound,  and  whose  denominator  is  the  num- 
ber of  vibrations  of  the  lower  or  graver  sound,  or  by  any  fraction 
equal  to  this. 

In  examining  the  relation  of  each  two  successive  numbers  in 
the  foregoing  series,  we  find  three  different  ratios.  Thus, 


27  :  24,  36  :  32,  and  45  :  40,  is  each  as    9 


8. 


30  :  27 and  40  :  36, 10       9. 

32  :  30 and  48  :  45, 16     15. 

Therefore,  of  the  seven  successive  intervals,  in  the  diatonic 
scale,  there  are  three  equal  to  •§,  two  equal  J^,  and  two  others 
equal  to  -J-f .  Each  of  the  first  five  is  called  a  tone ;  each  of  the 
last  two  is  called  a  semitone. 

323.  Repetition  of  the  Scale.— The  eighth  note  of  the 
scale  so  much  resembles  the  first  in  sound,  that  it  is  regarded  as  a 
repetition  of  it,  and  called  by  the  same  name.     Beginning,  there- 
fore, with  the  half  string,  where  the  former  series  closed,  let  us 
consider  the  sound  of  that  as  the  fundamental,  and  take  f  of  it 
for  the  second,  £  of  it  for  the  third,  &c.;  we  then  close  a  second 
series  of  notes  on  the  quarter-string,  whose  sound  is  also  con- 
sidered a  repetition  of  the  former  fundamental.     Each  fraction 
of  the  string  used  in  the  second  scale  is  obviously  half  of  the  cor- 
responding fraction  of  the  whole  string,    and  therefore  its  note  an 
octave  above  the  note  of  that.    This  process  may  be  repeated  in- 
definitely, giving  the  second  octave,   thiid  octave,   &c.    Ten  or 
eleven  octaves  comprehend  all  sounds  appreciable  by  the  human 
ear ;  the  vibrations  of  the  extreme  'notes  of  this  entire  range  have 
the  ratio  of  1  :  210,  or  1  :  211 ;  that  is,  1  :  1024,  or  1  :  2048.    Hence, 
if  16  vibrations  per  second  produce  the  lowest  appreciable  note, 
the  highest  varies  from  16,000  to  33,000.     It  was  ascertained  by 
Dr.  Wollaston  that  the  highest  limit  is  different  for  different  ears  ; 
so  that  when  one  person  complains  of  the  piercing  shrillness  of  a 
sound,  another  maintains  that  there  is  no  sound  at  all.      The 
lowest  limit  is  indefinite  for  a  different  reason ;  the  sounds  are 
heard  by  all,  but  some  will  recognize  them  as  low  musical  tones, 
while  others  only  perceive  a  rattling  or  fluttering  noise.    Few 
musical  instruments  comprehend  more  than  six  octaves,  and  the 
human  voice  has  only  from  one  to  three,  the  male  voice  being  in 
pitch  an  octave  lower  than  the  female. 

324.  Modes  of  Naming  the  Notes. — There  is  one  system 


THE    CHROMATIC    SCALE.  213 

of  names  for  the  notes  of  the  scale,  which  is  fixed,  and  another 
which  is  movable.  The  first  is  by  the  seven  letters,  A,  B,  C,  D, 
E,  F,  G.  The  notes  of  the  second  octave  are  expressed  by  the  same 
letters,  in  some  way  distinguished  from  the  former.  The  best 
method  is  to  write  by  the  side  of  the  letter  the  numeral  expressing 
that  index  of  2,  which  corresponds  to  the  octave:  as  A%,  A^, 
&c.,  in  the  octaves  above  ;  A^,  A%,  in  those  below. 

Any  note  may  be  designated  as  C,  but  this  letter  is  usually 
assigned  to  that  note  which  is  due  to  256  vibrations  per  second, 
or  middle  C  of  the  piano-forte. 

The  second  mode  of  designation  is  by  the  syllables,  do,  re,  mi, 
fa,  sol,  la,  si.  These  express  merely  the  relations  of  notes  to  each 
other,  do  always  being  the  fundamental,  re  its  second,  mi  its  third, 
&c.  In  the  natural  scale,  do  is  on  the  letter  C,  re  on  D,  &c.;  but 
by  the  aid  of  interpolated  notes,  the  scale  of  syllables  may  be 
transferred,  so  as  to  begin  successively  with  every  letter  of  the 
fixed  scale. 

325.  Tho  Chromatic  Scale. — Let  the  notes  of  the  diatonic 
scale  be  represented  (Fig.  202)  by  the  horizontal  lines,  C,  D,  &c.; 
the  distance  from  C  to  D  being  a  tone,  from  D  to  E  a 
tone,  E  to  F  a  semitone,  &c.     It  will  be  observed  that 
the  fundamental,  C,  is  so  situated  that  there  are  two 
whole  tones  above  it,  before  a  semitone  occurs,  and  then 
three  whole  tones  before  the  next  semitone.     C  is  there- 
fore  the  letter  to  be  called  by  the  syllable  do,  in  order  to 
bring  the  first  semitone  between  the  3d  and  4th,  and 
the  other  semitone  between  the  7th  and  8th,  as  the 
figure  represents  them.     Now,  that  we  may  be  able  to 
transfer  the  scale  of  relations  to  every  part  of  the  fixed 
scale  (which  is  necessary,  in  order  to  vary  the  character 
of  music,  without  throwing  it  beyond  the  reach  of  the 
voice),  the  whole  tones  are  bisected,  and  two  semitone 
intervals  occupy  the  place  of  each.     The  dotted  lines 
in  the  figure  show  the  places  of  the  interpolated  notes, 
which,  with  the  original  notes  of  the  diatonic  scale,  di-       - 
vide  the  whole  into  a  series  of  semitones.   This  is  called 
the  chromatic  scale.     The  interpolated  note  between  C 
and  D  is  written  C$  (C  sharp),  or  D  b  (D  flat),  and  so 
of  the  others.     As  the  whole  tones  lie  in  groups  of  twos 
and  threes,  so  the  new  notes  inserted  are  grouped  in  the 
same  way.    This  explains  the  arrangement  of  the  black 
keys  by  twos  and  threes  alternately  in  the  key-board  of  the  organ 
and  piano-forte.     The  white  keys  compose  the  diatonic  scale,  the 
white  and  black  keys  together,  the  chromatic  scale.     It  is  obvious 


214  ACOUSTICS. 

that  on  the  chromatic  scale  any  one  of  the  twelve  notes  which 
compose  it  may  become  do,  or  the  fundamental  note,  since  the  re- 
quired series,  2  tones,  1  semitone,  3  tones,  1  semitone,  can  be 
arranged  to  succeed  each  other,  at  whatever  note  we  begin  the 
reckoning.  This  change,  by  which  the  fundamental  note  is  made 
to  fall  on  different  letters,  is  called  the  transposition  of  the  scale. 

326.  Chords  and  Discords. — When  two  or  more  sounds, 
meeting  the  ear  at  once,  form  a  combination  which  is  agreeable, 
it  is  called  a  chord ;  if  disagreeable,  a  discord.  The  disagreeable 
quality  of  a  discord,  if  attended  to,  will  be  perceived  to  consist  in 
a  certain  roughness  or  harshness,  however  smooth  and  pure  the 
simple  sounds  which  are  combined.  On  examining  the  combina- 
tions, it  will  be  found  that  if  the  vibrations  of  two  sounds  are  in 
some  very  simple  relations,  as  1  :  2, 1  :  3,  2  :  3,  3  :  4,  &c.,  they  pro- 
duce a  chord ;  and  the  lower  the  terms  of  the  ratio,  the  more  per- 
fect the  chord.  On  the  other  hand,  if  the  numbers  necessary  to 
express  the  relations  of  the  sounds  are  large,  as  8  :  9,  or  15  :  16,  a 
discord  is  produced.  It  appears  that  concordant  sounds  have  fre- 
quent coincidences  of  vibrations.  If,  in  two  sounds,  there  is  coin- 
cidence at  every  vibration  of  each,  then  the  pitch  is  the  same,  and 
the  combination  is  called  unison.  If  every  vibration  of  one  coin- 
cides with  every  alternate  vibration  of  the  other,  the  ratio  is  1 :  2, 
and  the  chord  is  the  octave,  the  most  perfect  possible.  The  fifth  is 
the  next  most  perfect  chord,  where  every  second  vibration  of  the 
lower  meets  every  third  of  the  higher,  2  :  3.  The  fourth,  3  :  4,  the 
major  third,  4  :  5,  the  minor  third,  5  :  6,  and  the  sixth,  3  :  5,  are 
reckoned  among  chords  ;  while  the  second,  8  :  9,  and  the  seventh, 
8  :  15,  are  harsh  discords.  What  is  called  the  common  chord  con- 
sists of  the  1st,  3d,  and  5th,  combined,  and  is  far  more  used  in 
music  than  any  other.  Harmony  consists  of  a  succession  of  chords, 
or  rather,  of  such  a  succession  of  combined  sounds  as  is  pleasing 
to  the  ear  ;  for  discords  are  employed  in  musical  composition, 
their  use  being  limited  by  special  rules.  Many  combinations, 
which  would  be  too  disagreeable  for  the  ear  to  dwell  upon,  or  to 
finish  a  musical  period,  are  yet  quite  necessary  to  produce  the  best 
effect ;  and  without  the  relief  which  they  give,  perfect  harmony, 
if  long  continued,  would  satiate. 

327.  Temperament. — This  is  a  term  applied  to  the  small 
errors  introduced  into  the  notes,  in  tuning  an  instrument  of  fixed 
keys,  in  order  to  adapt  the  notes  equally  to  the  several  scales.  If 
the  tones  were  all  equal,  and  if  semitones  were  truly  half  tones, 
no  such  adjustment  of  notes  would  be  needed ;  they  would  all 
be  exactly  correct  for  every  scale.  Representing  the  notes  in 


TEMPERAMENT.  215 

the  scale  whose  fundamental  is  C  by  the  numbers  in  Art.  321,  we 
have, 

C,    D,  E,   F,    G,   A,   B  C9,  D2,  $„  &c. 

24,  27,  30,  32,  36,  40,  45,  48,   54,    60,    &c. 

Now  suppose  we  wish  to  make  />,  instead  of  C,  our  key-note  ; 
then  it  is  obvious  that  E  will  not  be  exactly  correct  for  the  second 
on  the  new  scale.  For  the  fundamental  to  its  second  is  as  8  to  9  ; 
and  8  :  9  ::  27  :  30.375,  instead  of  30.  Therefore,  if  D  is  the  key- 
note, we  must  have  a  new  E,  slightly  above  the  E  of  the  original 
scale.  So  we  find  that  A,  represented  by  40,  will  not  serve  to  be 
the  5th  in  the  new  scale ;  since  2  :  3  : :  27  :  40.5,  which  is  a  little 
higher  than  A  (=  40).  After  adding  these  and  other  new  notes, 
to  render  the  intervals  all  exactly  right  for  the  new  key  of  D,  if 
we  proceed  in  the  same  manner,  and  make  E  (=  30)  our  key-note, 
and  obtain  its  second,  third,  &c.,  exactly,  we  shall  find  some  of 
them  differing  a  little,  both  from  those  of  the  key  of  C,  and  also 
of  the  key  of  D.  Using  in  this  way  all  the  twelve  notes  of  the 
chromatic  scale  in  succession  for  the  fundamental,  it  appears  that 
several  different  E9s,  F's,  G's,  &c.,  are  required,  in  order  to  make 
each  scale  perfect.  In  instruments,  whose  sounds  cannot  be  mod- 
ified by  the  performer,  like  the  organ  and  piano-forte,  as  it  is  con- 
sidered impossible  to  insert  all  the  pipes  or  strings  necessary  to 
render  every  scale  perfect,  such  an  adjustment  is  made  as  to  dis- 
tribute these  errors  equally  among  all  the  scales.  For  example,  E 
is  not  made  a  perfect  third  for  the  key  of  (7,  lest  it  should  be  too 
imperfect  for  a  second  in  the  key  of  D,  and  for  its  appropriate 
place  in  other  scales.  It  is  this  equal  distribution  of  errors  among 
the  several  scales  which  is  called  temperament.  The  errors,  when 
thus  distributed,  are  too  small  to  be  observed  by  most  persons ; 
whereas,  if  an  instrument  was  tuned  perfectly  for  any  one  scale, 
all  others  would  be  intolerable. 

The  word  temperament,  as  above  explained,  has  no  application 
except  to  instruments  of  fixed  keys,  as  the  organ  and  piano-forte ; 
for,  where  the  performer  can  control  and  modify  the  notes  as  he  is 
playing,  he  can  make  every  key  perfect,  and  then  there  are  no 
errors  to  be  distributed.  The  flute-player  can  roll  the  flute 
slightly,  and  thus  humor  the  sound,  so  as  to  cause  the  same 
fingering  to  give  a  precisely  correct  second  for  one  scale,  a  correct 
third  for  another,  and  so  on.  The  player  on  the  violin  does  the 
same,  by  touching  the  string  in  points  slightly  different.  The  or- 
gans of  the  voice,  especially,  can  be  adjusted  to  make  the  intervals 
perfect  on  every  scale.  In  these  cases  there  is  no  tempering,  or 
dividing  of  errors  among  different  scales,  but  a  perfect  adjustment 
to  each  scale,  by  which  all  error  is  avoided. 


216  ACOUSTICS. 

328.  Harmonics. — The  fact  has  been  mentioned  that  a  string, 
or  a  column  of  air,  may  vibrate  in  parts,  even  while  vibrating  as 
a  whole.      It  only  remains  to  show  the  musical  relations  of  the 
sounds  thus  produced.     When  a  string  vibrates  in  parts,  it  divides 
into  halves,  thirds,  fourths,  or  other  aliqwt  parts.     Now,  a  half- 
string  produces  an  octave  above  the  whole,  making  the  most  per- 
fect chord  with  it.     The  third  of  a  string  being  two-thirds  of  the 
half-string,  produces  the  fifth  above  the  octave,  a  very  perfect 
chord.     The  quarter-string  gives  the  second  octave  ;  the  fifth  part 
of  it,  being  £  of  the  quarter,  gives  the  major  third  above  the 
second  octave  ;  and  the  sixth  part,  being  f  of  the  quarter,  gives 
the  Jifth  above  the  second  octave.    Thus,  all  the  simpler  divisions, 
which  are  the  ones  most  likely  to  occur,  are  such  as  produce  the 
best  chords ;  and  it  is  for  this  reason  that  the  sounds  are  called 
harmonics.     The  same  is  true  of  air-columns  and  bells.     The 
^Eolian  harp  furnishes  a  beautiful  example  of  the  harmonics  of  a 
string.     Two  or  more  fine  smooth  cords  are  fastened  upon  a  box, 
and  tuned,  at  suitable  intervals,  like  the  strings  of  a  violin;  and 
the  box  is  placed  in  a  narrow  opening,  where  a  current  of  air 
passes.    Each  string  at  different  times,  according  to  the  intensity 
of  the  breeze,  will  emit  a  pure  musical  note ;  and,  with  every 
change,  will  divide  itself  in  a  new  mode,  and  give  another  pitch, 
while  it  will  frequently  happen  that  the  vibrations  of  different 
divisions  will  coexist,  and  their  harmonic  sounds  mingle  with  each 
other. 

329.  Overtones. — But  the  parts  into  which  a  sounding  body 
divides  do  not  always  harmonize  with  the  whole.     For  instance,  \ 
or  ^  of  a  string  is  discordant  with  the  fundamental.     The  word 
harmonics  is  not,  therefore,  applicable  except  to  a  very  few  of  the 
many  possible  sounds  which  a  body  may  produce.     The  word 
overtone  is  used  to  express  in  general  any  sound  whatever,  given 
by  a  part  of  a  sounding  body.     A  string  may  furnish  20  or  30 
overtones,  but  only  a  small  number  of  them  would  be  harmonics. 

The  presence  of  these  overtones  may  be  determined  by 
means  of  the  resonator  devised  by  Helmholtz  and  modified  by 
Konig,  which  can  be  adjusted  to  respond  to  a  great  variety  of 
notes  ;  if  on  drawing  out  the  cylinder  a  tone  is  produced,  it  must 
be  that  the  same  tone  exists  in  the  compound  sound  under  inves- 
tigation. 

330.  Timbre,  or  Quality  of  Tone. — Even  when  the  pitch 
of  two  sounding  bodies  is  the  same,  the  ear  almost  always  distin- 
guishes one  sound  from  the  other   by  certain  qualities  of  tone 
peculiar  to  each.     Thus,  if  the  same  letter  be  sounded  by  a  flute 


COMMUNICATION    OF    VIBRATIONS.  21? 

and  the  string  of  a  piano,  each  note  is  easily  distinguished  from 
the  other.  Two  church-bells  may  be  upon  the  same  key,  and  yet 
one  be  agreeable,  and  the  other  harsh  to  the  ear. 

As  a  result  of  his  researches  Helmholtz  decides  that  the  timbre 
is  determined  by  the  overtones  which  accompany  the  primary 
tones.  If  these  overtones  could  be  eliminated,  leaving  only  the 
pure,  simple  tones,  a  note  of  given  pitch  sounded  by  a  flute  would 
not  be  distinguishable  from  a  note  of  the  same  pitch  sounded  by 
a  violin. 

A  long  monochord  can,  by  varying  the  mode  of  exciting  the 
vibrations,  be  made  to  yield  a  great  variety  of  sounds,  while  there 
is  perceived  in  them  all  the  same  fundamental  undertone  which 
determines  the  pitch.  If  the  string  be  struck  at  the  middle,  then 
no  node  can  be  formed  at  that  point ;  hence,  the  mixed  sound  will 
contain  no  overtones  of  the  £,  £,  £,  -J-,  TV?  or  other  even  aliquot 
parts  of  the  string ;  for  all  such  would  require  a  node  at  the 
middle.  But  if  struck  at  one-third  of  its  length  from  the  end, 
then  the  overtones,  £,  J,  &c.,  may  exist,  but  not  those  of  J,  -£,  -J-, 
or  any  other  parts  whose  node  would  fall  at  J  of  the  length  from 
the  end. 

For  reasons  which  are  mostly  unknown,  some  sounding  bodies 
have  their  fundamental  accompanied  by  harmonic  overtones,  and 
others  by  overtones  which  are  discordant.  And  this  is  one  cause 
of  the  agreeable,  or  unpleasant,  quality  of  the  sounds  of  different 
bodies. 

331.  Communication  of  Vibrations. — The  acoustic  vibra- 
tions of  one  body  are  readily  communicated  to  others,  which  are 
near  or  in  contact.  We  have  already  noticed  that  the  vibrations 
of  a  reed  will  excite  those  of  a  column  of  air  in  a  pipe.  If  two 
strings,  which  are  adapted  to  vibrate  alike,  are  fastened  on  the 
same  box,  and  one  of  them  is  made  to  sound,  the  other  will  sound 
also  more  or  less  loudly,  according  to  the  intimacy  of  their  connec- 
tion. The  vibrations  are  communicated  partly  through  the  air, 
and  partly  through  the  materials  of  the  box.  So,  if  a  loud  sound 
is  uttered  near  a  piano-forte,  several  strings  will  be  thrown  into 
vibration,  whose  notes  are  heard  after  the  voice  ceases.  The  no- 
ticeable fact  in  all  such  experiments  is,  that  the  vibrations  thus 
communicated  from  one  body  to  another  cause  sounds  which  har- 
monize with  each  other,  and  with  the  original  sound.  For  the 
rate  of  vibration  will  either  be  identical,  or  have  those  simple  re- 
lations which  are  expressed  by  the  smallest  numbers.  Let  a  per- 
son hold  a  pneumatic  receiver  or  a  large  tumbler  before  him,  and 
utter  at  the  mouth  of  it  several  sounds  of  different  pitch  ;  and  he 
will  probably  find  some  one  pitch  which  will  be  distinctly  rein- 


218  ACOUSTICS. 

forced  by  the  vessel.  That  particular  note,  which  the  receiver  by 
its  size  and  form  is  adapted  to  produce,  will  not  be  called  forth  by 
a  sound  that  would  be  discordant  with  it.  The  melodeon,  ser- 
aphine,  and  instruments  of  like  character,  owe  their  full  and  bril- 
liant notes  to  reeds,  each  of  which  has  its  cavity  of  air  adapted  to 
vibrate  in  unison  with  it.  It  sometimes  happens  that  the  second 
body,  vibrating  as  a  whole,  would  not  harmonize  with  the  first,  and 
yet  will  give  the  same  note  by  some  mode  of  division.  Thus  it  is 
that  all  the  various  sounds  of  the  monochord,  and  of  the  strings 
of  the  viol,  are  reinforced  by  the  case  of  thin  wood  upon  which 
they  are  stretched.  The  plates  of  wood  divide  by  nodal  lines  into 
some  new  arrangement  of  ventral  segments  for  every  new  sound 
emitted  by  the  string.  In  like  manner,  the  pitch  of  the  tuning- 
fork,  and  all  the  rapid  notes  of  a  music-box,  are  rendered  loud  and 
full  by  the  table,  in  contact  with  which  they  are  brought.  The 
extended  material  of  the  table  is  capable  of  division  into  a  great 
variety  of  forms,  and  will  always  give  a  sound  in  unison  with  the 
instrument  which  touches  it. 

332.  One  System  of  Vibrations  Controlling  Another.— 
If  two  sounding  bodies  are  nearly,  but  not  precisely  on  the  same 
key,  they  will  sometimes,  when  brought  into  close  contact,  be 
made  to  harmonize  perfectly.     The  vibrations  of  the  more  power- 
ful will  be  communicated  to  the  other,  and  control  its  movements 
so  that  the  discordance,  which  they  produce  when  a  few  inches 
apart,  will  cease,  and  concord  will  ensue.     Two  diapason  pipes  of 
an  organ,  tuned  a  quarter-tone  or  even  a  semitone  from  unison,  so 
as  to  jar  disagreeably  upon  the  ear,  when  one  inch  or  more  asun- 
der, will  be  in  perfect  unison,  if  they  are  in  contact  through  their 
whole  length.     Even  the  slow  oscillations  of  two  watches  will  in- 
fluence each  other ;  if  one  gains  on  the  other  only  a  few  beats 
in  an  hour,  then,  if  they  are  placed  side  by  side  on  the  same  board; 
they  will  beat  precisely  together. 

333.  Crispations  of  Fluids. — Among  the  numerous  acous- 
tic experiments  illustrating  the  communication  of  vibrations,  none 
are  more  beautiful  than  those  in  which  the  vibrations  of  glass  rods 
are  conveyed  to  the  surface  of  a  fluid.     Let  a  very  shallow  pan  of 
glass  or  metal  be  attached  to  the  middle  of  a  thin  bar  of  wood, 
three  or  four  feet  long,  and  resting  near  its  ends  on  two  fixed 
bridges ;  let  water  be  placed  in  the  pan,  and  a  long  glass  rod 
standing  in  it,  or  on  the  wood,  be  vibrated  longitudinally,  by 
drawing  the  moistened  fingers  down  upon  it ;  the  liquid  immedi- 
ately shows  that  the  vibrations  are  communicated  to  it.    The  sur- 
face is  covered  with  a  regular  arrangement  of  heaps,  called  crispa- 
tions,  which  vary  in  size  with  the  pitch  of  sound,  which  is  produced 


INTERFERENCE.  £1!) 

by  the  same  vibration.  If  the  pitch  is  higher,  4;hey  are  smaller, 
and  may  be  readily  varied  from  three  or  four  inches  in  diameter 
to  the  fineness  of  the  teeth  of  a  file.  Crispations  of  the  same 
character  are  also  formed  in  clusters  on  the  water  in  a  large  tum- 
bler or  glass  receiver,  when  the  finger  is  drawn  along  its  edge  ; 
every  ventral  segment  of  the  glass  produces  a  group  of  hillocks  by 
the  side  of  it  on  the  surface  of  the  water. 

334.  Interference  of  Waves  of  Sound.—  Whenever  two 
sounds  are  moving  through  the  air,  every  particle  will,  at  a  given 
instant,  have  a  motion  which  is  the  resultant  of  the  two  motions 
which  it  would  have  had  if  the  sounds  were  separate.  These  mo- 
tions may  conspire,  or  they  may  oppose  each  other.  The  word 
interference  is  used  in  scientific  language  to  express  the  resultant 
effect,  whatever  it  may  be. 

If  two  sound  waves  of  equal  intensity  and  of  the  same  length 
move  together  so  that  any  phase  of  one  is  coincident  with  the  like 
phase  of  the  other  the  resultant  sound  has  greater  intensity  than 
either  of  the  components  ;  if,  however,  any  phase  of  one  is  coinci- 
dent with  the  opposite  phase  of  the  other,  entire  extinction  of  the 
sound  results,  and  we  have  silence.  To  illustrate  this  experi- 
mentally, take  two  pieces  of  tubing  and  bend  them  into  the  form 
shown  in  Fig.  203,  the  branch  A  being  large  enough  to  slide  over 

the  other,  and  insert  a 

FIG-  m  whistle  at  D.    The  sound 

waves  travel  to  the  ear 
placed  at  C  by  two  differ- 
ent  routes,  starting  in 
the  same  phase  at  the 
point  D.  If  the  branches 
A  and  B  are  of  equal 

length,  or  differ  by  one  or  more  whole  wave  lengths,  the  waves 
will  meet  at  C  in  the  same  phase  and  produce  a  sound  of  greater 
intensity  than  that  of  either  alone  ;  but  if  the  branch  A  be 
drawn  out  or  pushed  in  till  the  route  A  differs  in  length  from 
B  by  half  a  wave  length,  or  by  any  odd  multiple  of  half  a 
wave  length,  opposite  phases  will  meet  at  C  and  destroy  each 
other,  producing  silence.  A  much  more  simple  experiment  to 
show  the  same  effect  is  the  following  :  The  two  prongs  of  a  tuning- 
fork  always  vibrate  in  opposite  directions,  one  producing  a  con- 
densation in  the  direction  in  which  the  other  produces  rarefaction, 
thus  destroying  each  other's  effect  by  interference,  and  hence  the 
almost  total  absence  of  sound  when  the  fork  is  held  free  in  the 
hand.  In  such  case,  if  the  sound  waves  from  one  prong  be  inter- 
cepted by  slipping  over  it  without  contact  a  paper  cylinder,  the 


^ 

j\B 

^ 


220  ACOUSTICS. 

sound  is  augmented.  Hold  a  vibrating  fork  so  that  either  the 
back  face  of  a  prong,  or  the  side  faces  of  the  two  prongs,  are 
parallel  to  the  ear,  and  the  sound  will  be  distinctly  audible  ;  turn 
the  fork  about  its  axis  45°  from  either  of  these  positions,  and 
silence  results.  During  one  entire  rotation  of  the  fork  there  will 
be  four  positions  of  maximum  intensity  and  four  other  positions 
of  total  extinction  of  the  sound.  If  the  fork  be  rotated  over  the 
mouth  of  a  resonating  jar,  the  effect  is  much  more  striking. 

The  beats,  which  are  frequently  heard  ,in  listening  to  two 
sounds,  indicate  the  points  of  maximum  condensation  produced 
by  the  union  of  the  condensed  parts  of  both  systems  of  waves. 
And  the  sounds  are  considered  discordant  when  these  beats  are 
just  so  frequent  as  to  produce  a  disagreeable  fluttering  or  rattling. 
If  too  near  or  too  far  apart  for  this,  they  are  regarded  practically 
as  concordant.  And  when  the  beats  are  too  close  to  be  perceived 
separately,  yet  the  peculiar  adjustment  of  condensations  of  one 
system  with  those  of  the  other,  according  as  one  wave  measures 
two,  or  two  waves  measure  three,  or  four  measure  five,  &c.,  is  at 
once  distinguished  by  the  ear,  and  recognized  as  the  chord  of  the 
octave,  the  fifth,  the  third,  &c.  When  a  sound  and  its  octave  are 
advancing  together,  there  are  instants  in  which  any  given  particle 
of  air  is  impressed  with  two  opposite  motions,  and  other  alternate 
moments  when  both  motions  are  in  the  same  direction.  For  the 
waves  of  the  highest  sound  are  half  as  long  as  those  of  the  lowest ; 
hence,  while  every  second  condensation  of  the  former  coincides 
with  every  condensation  of  the  latter,  the  alternate  ones  of  the 
former  must  be  at  the  points  of  greatest  rarefaction  of  the  latter  ; 
and  this  cannot  occur  without  opposite  movements  of  the  par- 
ticles. 

335.  Number  and  Length  of  Waves  for  Each  Note. — 
Though  the  vibrations  of  any  musical  note  are  too  rapid  to  be 
counted,  yet  the  number  may  be  ascertained  in  several  ways. 

If  an  elastic  slip  of  metal  be  clamped  at  one  end  so  that  the 
other  end  may  rest  against  a  toothed  wheel,  and  the  wheel  be  re- 
volved with  different  velocities,  musical  notes  of  different  pitch 
will  be  produced.  If  a  uniform  velocity  be  maintained  for  a  given 
time,  sixty  seconds  for  instance,  and  the  number  of  revolutions  of 
the  toothed  wheel  be  read  from  an  indicator  suitably  connected 
with  the  wheel,  then  the  number  of  teeth  upon  the  wheel  multi- 
plied by  the  number  of  revolutions  gives  the  number  of  impulses 
communicated  to  the  air  in  the  given  time  ;  this  product,  divided 
by  the  seconds  in  the  time,  sixty  in  the  case  supposed,  gives  the 
number  of  vibrations  per  second.  The  instrument  is  called 
Savart's  Wheel. 


LENGTH    OF    WAVE.  221 

Another  method  of  determining  the  number  of  vibrations  is  by 
means  of  the  siren,  invented  by  De  La  Tour.  In  this  instrument 
the  pulses  are  produced  by  puffs  of  air,  in  rapid  succession,  caused 
by  revolving  a  disc,  perforated  around  its  circumference  by 
numerous  holes  which  pass  in  front  of  an  air  jet.  The  calcu- 
lation is  similar  to  that  given  for  Savart's  wheel. 

In  the  method  given  above  it  is  difficult  in  practice  to  main- 
tain a  constant  velocity.  A  graphic  representation  of  the  vibra- 
tions, devised  by  Duhamel,  is  without  this  objection.  Without 
giving  details  of  the  construction,  the  principle  of  the  method 
may  be  given  as  follows : 

Support  the  vibrating  rod — a  tuning-fork  for  example— above 
a  table,  as  in  Fig.  204;  to  one  prong  of  the  fork  attach  a  fine  steel 
point  or  style,  which 

shall      very      lightly  Fl<*.  204. 

touch  the  strip  of 
paper  D  G,  which  has 
been  coated  with  a 
film  of  lamp  black ; 

place  at  /  an  electric  _______^======a=__==_ 

style,  which  is  in  con- 
nection with  a  pendulum  beating  seconds,  which  shall  make  a 
dot  upon  the  paper  at  each  beat  of  the  pendulum.  If  the  paper 
be  moved  in  the  direction  D  G  while  the  fork  is  kept  vibrating, 
an  undulating  line  will  be  traced  upon  the  film,  and  the  number 
of  undulations  between  any  two  consecutive  second's  dots,  as  b  a, 
gives  the  number  of  vibrations  of  the  fork  per  second. 

In  these  ways  it  is  ascertained  that  the  numbers  corresponding 
to  the  letters  of  the  scale  are  the  following: 

(    (7,     A     E,      F,      G,      A,      B,     Cz, 
\  128,  144,  160,  170f,  192,  213|,  240,  256, 

The  highest  note  of  the  above  series,  (72,  256,  is  the  middle  C  of 
the  piano-forte. 

There  is  not,  however,  a  perfect  agreement  of  pitch  in  differ- 
ent countries,  and  among  different  classes  of  musicians.  Accord- 
ingly, (7,  which  is  given  above  as  corresponding  to  128  vibrations 
per  second,  has  several  values,  varying  from  127  to  131. 

To  find  the  length  of  acoustic  waves  for  any  given  pitch,  we 
have  only  to  divide  the  velocity  of  sound  in  one  second  by  the 
number  of  vibrations  which  reach  the  ear  in  the  same  length  of 
time.  For  example,  at  the  temperature  of  60°  F.  sound  travels  in 
air  1121  feet  per  second  ;  therefore  the  length  of  waves  of  middle 
C  on  the  piano  =  1121  -~  256  —  4.4  feet  nearly,  a  wave  length 
being  the  distance  through  which  sound  moves  during  one  vibra* 


ACOUSTICS. 

tion  (Par.  285).  The  waves  of  the  lowest  musical  note  are  about 
70  feet  long ;  and  of  the  highest,  less  than  half  an  inch. 

336.  Doppler's   Principle. — Thus  far  the  hearer  and  the 
source  of  sound  have  been  supposed  not  to  change  their  relative 
positions.    When  a  sounding  body  approaches  the  ear,  the  tone 
perceived  is  higher  than  that  due  to  the  number  of  vibrations  per 
second,  since  more  vibrations  per  second  reach  the  ear  than  if  the 
body  had  remained  at  rest  with  respect  to  the  hearer.     Suppose 
the  sound  to  be  middle  C,  and  the  sounding  body  and  the  hearer 
to  remain  relatively  stationary,  then  256  vibrations  per  second 
will  be  communicated  to  the  ear:  if  now  the  sounding  body  ap- 
proach at  the  rate  of  66  feet  per  second  there  will  be  perceived,  in 

fifi 
addition  to  the  256  vibrations,  j-j  =  15  vibrations  per  second, 

and  the  pitch  will  be  that  due  to  271  vibrations  per  second  in- 
stead of  256.  If  the  sounding  body  recede  from  the  hearer  the 
opposite  effect  will  be  produced. 

337.  Acoustic  Vibrations  Visibly  Projected.— The  vi- 
brations of  heavy  tuning-forks  can  be  magnified  and  rendered 
distinctly  visible  to  an  audience  by  projecting  them  on  a  screen. 
The  fork  being  constructed  with  a  small  metallic  mirror  attached 
near  the  end  of  one  prong,  a  sunbeam  reflected  from  the  mirror 
will  exhibit  all  the  movements  of  the  fork  greatly  enlarged  on  a 
distant  wall ;  and  if  the  fork  is  turned  on  its  axis,  the  luminous 
projection  will  take  the  form  of  a  waving  line.    And  by  the  use  of 
two  forks,  all  the  phenomena  of  interference  may  be  rendered  as 
distinct  to  the  eye  as  they  are  to  the  ear. 

338.  The  Telephone. — This  instrument  for  reproduction 
of  sound  at  a  distance  by  means  of  electric  currents  is  shown  in 
section  in  Fig.  205  in  which  a  a'  is  a  disc  or  diaphragm  of  thin 

soft    iron,    the    circum- 

Fio-  205.  ference  of  which  is  firmly 

clamped  between  the 
mouth  guard  m  m'  and 
the  case  n  ri,  upon  the 
centre  of  which  the 
sound  waves  from  the 
mouth  impinge,  as  at  E, 
and  communicate  to  it 
vibrations  corresponding  to  the  simple  or  composite  sounds  ut- 
tered. These  vibrations  of  the  disc  cause  a  continual  variation 
in  the  distance  of  the  disc  from  the  end  of  a  bar  magnet  I. 
Around  the  end  of  the  magnet  nearest  to  the  diaphragm  a  a'  is  a 


THE    TELEPHONE.  22o 

helix  c  c'  of  fine  insulated  copper  wire,  the  ends  of  which  are  con- 
nected with  binding  posts  dy  d',  From  these  posts  are  carried 
wires  to  another  precisely  similar  instrument  at  the  station  with 
which  communication  is  to  be  held.  When  a  word  is  spoken  into 
the  instrument  at  E,  the  vibrations  communicated  to  the  disc  a  a' 
cause  variations  in  the  magnetic  intensity  of  the  bar  I,  and  these 
variations  cause  electric  currents  to  flow  in  the  helix  c  c'  and 
thence  through  the  connecting  wires  to  the  helix  in  the  instru- 
ment held  to  the  ear  of  the  listener,  and  these  currents  in  the 
last  named  helix  produce  variations  in  the  magnet  of  the  receiv- 
ing instrument,  causing  precisely  the  same  vibrations  in  its  dia- 
phragm as  were  originally  set  up  in  the  first.  The  vibrations  of 
the  diaphragm  are  transmitted  through  the  air  to  the  ear ;  and 
though  no  sound  has  been  transmitted  from  one  station  to  the 
other,  the  words  spoken  into  one  instrument  are  distinctly  de- 
livered by  the  o^her.  The  sound  vibrations  are  the  cause  of 
electric  currents,  as  is  explained  elsewhere,  and  these  in  turn 
finally  produce  sound  vibrations  again. 

To  such  perfection  of  action  have  these  instruments  been 
brought,  that  not  only  can  the  spoken  words  be  heard,  but  the 
peculiar  characteristics  of  voice  are  so  faithfully  reproduced  that 
by  these  the  speaker  may  be  recognized. 


PART    Y, 

OPTICS. 

CHAPTER    I. 

MOTION   AND  INTENSITY   OF   LIGHT. 

339.  Definitions. — Light  is  supposed  to  consist  of  exceed- 
ingly minute  and  rapid  vibrations  in  a  medium  or  ether  which 
fills  space ;  which  vibrations,  on  reaching  the  retina  of  the  eye,, 
cause  vision,  as  the  vibrations  of  the  air  cause  hearing,  when  they 
impinge  on  the  tympanum  of  the  ear,  and  as  thermal  vibrations 
produce  a  sensation  of  warmth,  when  they  fall  on  the  skin ;  the 
difference  between  light  and  heat  is  solely  a  difference  in  wave 
length,  waves  longer  than  those  of  the  extreme  red,  or  shorter 
than  the    extreme  violet,  producing  no  effect  upon   the  optic 
nerve. 

Bodies,  which  of  themselves  are  able  to  produce  vibrations  in 
the  ether  surrounding  them,  are  said  to  emit  light,  and  are  called 
self -luminous )  or  simply  luminous  ;  those,  which  only  reflect  light, 
are  called  non-luminous.  Most  bodies  are  of  the  latter  class.  A 
ray  of  light  is  a  line,  along  which  light  is  propagated  ;  a  beam  is 
made  up  of  many  parallel  rays ;  &  pencil  is  composed  of  rays  either 
diverging  or  converging  ;  and  is  not  unfrequently  applied  to  those 
which  are  parallel. 

A  substance,  through  which  light  is  transmitted,  is  called  a 
medium  ;  if  objects  are  clearly  seen  through  the  medium,  it  is 
called  transparent ;  if  seen  faintly,  semi-transparent ;  if  light  is 
discerned  through  a  medium,  but  not  the  objects  from  which  it 
comes,  it  is  called  translucent ;  substances  which  transmit  no  light 
are  called  opaque,  though  all  are  more  or  less  translucent  when 
cut  in  sufficiently  thin  laminae. 

340.  Light  Moves  in  Straight  Lines. — So  long  as  the 
medium  continues  uniform,  the  line  of  each  ray  is  perfectly 


THE    VELOCITY    OF    LIGHT. 

straight.  For  an  object  cannot  be  seen  through  a  bent  tube ; 
and  if  three  discs  have  each  a  small  aperture  through  it,  a  ray 
cannot  pass  through  the  three,  except  when  they  are  exactly  in  a 
straight  line.  The  shadow  which  is  projected  through  space 
from  an  opaque  body  proves  the  same  thing;  for  the  edges  of  the 
shadow,  taken  in  the  direction  of  the  rays,  are  all  straight  lines. 

From  every  point  of  a  luminous  surface  light  emanates  in  all 
possible  directions,  when  not  prevented  by  the  interposition  of 
an  opaque  body.  Thus,  a  candle  is  seen  by  night  at  the  distance 
of  one  or  two  miles ;  and  within  that  limit,  no  space  so  small  as 
the  pupil  of  the  eye  is  destitute  of  rays  from  the  candle.  A 
point  from  which  light  emanate's  is  called  a  radiant.  If  light 
from  a  radiant  falls  perpendicularly  on  a  circular  disk,  the  pencil 
is  a  cone;  if  on  a  square  disk,  it  is  a  square  pyramid,  &c.,  the 
illuminated  surface  in  each  case  being  the  base,  and  the  radiant 
the  vertex. 

341.  The  Velocity  of  Light. — It  has  been  ascertained  by 
several  independent  methods,  that  light  moves  at  the  rate  of  about 
186,300  miles  per  second. 

One  method  is  by  means  of  the  eclipses  of  Jupiter's  satellites. 
The  planet  Jupiter  is  attended  by  four  moons  which  revolve  about 
it  in  short  periods.  These  small  bodies  are  observed,  by  the  tele- 
scope, to  undergo  frequent  eclipses  by  falling  into  the  shadow 
which  the  planet  casts  in  a  direction  opposite  to  the  sun.  The 
exact  moment  when  the  satellite  passes  into  the  shadow,  or  comes 
out  of  it,  is  calculated  by  astronomers.  But  sometimes  the  earth 
and  Jupiter  are  on  the  same  side,  and  sometimes  on  opposite  sides 
of  the  sun  ;  consequently,  the  earth  is,  in  the  former  case,  the 
whole  diameter  of  its  orbit,  or  about  one  hundred  and  eighty-five 
millions  of  miles  nearer  to  Jupiter  than  in  the  latter.  Now  it  is 
found  by  observation,  that  an  eclipse  of  one  of  the  satellites  is 
seen  about  sixteen  minutes  and  a  half  sooner  when  the  earth  is 
nearest  to  Jupiter,  than  when  it  is  most  remote  from  it,  and  con- 
sequently, the  light  must  occupy  this  time  in  passing  through  the 
diameter  of  the  earth's  orbit,  and  must  therefore  travel  at  the  rate 
of  about  186,868  miles  per  second,  according  to  this  determina- 
tion. 

Another  method  of  estimating  the  velocity  of  light,  wholly 
independent  of  the  preceding,  is  derived  from  what  is  called  the 
aberration  of  the  fixed  stars.  The  apparent  place  of  a  fixed  star 
is  altered  by  the  motion  of  its  light  being  combined  with  the  mo- 
tion of  the  earth  in  its  orbit.  The  place  of  a  luminous  object  is 
determined  by  the  direction  in  which  its  light  meets  the  eye.  But 
the  direction  of  the  impulse  of  light  on  the  eye  is  modified  by  the 
15 


226 


OPTICS. 


motion  of  the  observer  himself,  and  the  object  appears  forward  of 
its  true  place.  The  stars,  for  this  reason,  appear  slightly  displaced 
in  the  direction  in  which  the  earth  is  moving ;  and  the  velocity 
of  the  earth  being  known,  that  of  light  may  be  computed  in  the 
same  manner  as  we  determine  one  component,  when  the  angles 
and  the  other  component  are  known. 

342.  Determination  of  the  Velocity  of  Light  by  Ex- 
periment.— The  velocity  of  light  has  been  determined  also  by 
experiment,  in  a  manner  somewhat  analogous  to  that  employed 
by  Wheatstone  for  ascertaining  the  velocity  of  electricity.  The 
method  adopted  by  Foucault  is  essentially  the  following  : 

Through  an  aperture  A,  in  a  shutter  (Fig.  206),  a  beam  of 
light  is  admitted,  which  passing  through  an  inclined  transparent 

FIG.  206. 


glass  mirror,  E,  and  through  a  lens  of  very  long  focus,  B,  falls 
upon  a  mirror  (7,  and  is  reflected  to  a  mirror  D ;  the  mirror  D 
again  reflects  the  beam  back  to  (7,  whence  it  is  returned  through 
the  lens  B  to  the  glass  mirror  E,  is  reflected,  and  finally  enters 
the  eye.  The  mirror  C  is  so  mounted  as  to  rotate  with  great 
velocity  upon  an  axis,  perpendicular  to  the  plane  of  the  ptiper  in 
the  case  supposed. 

If  the  mirror  0  rotate  slowly,  in  the  direction  of  the  arrow, 
the  beam  will  alternately  disappear  and  reappear  at  the  point  a  ; 
but  if  the  velocity  be  increased  to  30  or  more  revolutions  per 
second  the  impression  on  the  eye  becomes  persistent  and  the 
beam  is  seen  without  interruptions,  appearing  stationary  at  a.  If 
now  the  speed  of  the  mirror  C  be  increased  to  from  300  to  600 
revolutions  per  second,  the  change  of  position  of  the  mirror  C, 
while  light  is  passing  from  it  to  D  and  back  again,  is  sufficient  to 
return  the  reflected  beam  to  some  point  #,  the  distance  from  a 
depending  upon  the  velocity  of  rotation.  From  the  displacement 
at  b,  the  angular  motion  of  the  mirror  at  C,  while  the  beam  tra- 
verses the  distance  from  C  to  D  and  back  again,  can  be  determined ; 
and  knowing  the  rate  of  rotation,  this  fraction  of  one  turn  gives 


LOSS    OF    INTENSITY    BY    DISTANCE.  227 

the  time  which  the  light  required  to  traverse  double  the  distance 
C  I),  and  hence  its  velocity.  Such  is  an  outline  of  the  mode  of 
experimenting,  all  details  for  securing  precision  having  been 
omitted. 

By  this  method  the  velocity  given  in  Art.  341 — 186,300  miles 
per  second — was  determined  by  A.  A.  Michel  son,  U.  S.  N.  The 
distance  between  the  revolving  mirror  C  and  the  mirror  D  was 
2000  feet. 

343.  Loss  of  Intensity  by  Distance. — The  intensity  of 
light  varies  inversely  as  the  square  of  the  distance.     In  Fig.  207, 
suppose  light  to  radi- 
ate from  8,  through 

the  rectangle  A  C, 
and  fall  on  EG,  paral- 
lel to  AC.  AsSAE, 
S  B  F,  &c.,  are 
straight  lines,  the 
triangles,  S  A  B, 
S  E  F,  are  similar, 
as  also  the  rectangles,  A  C,  E  G ;  therefore,  A  C  :  E  G  : :  A  &  : 
E  F2  : :  S  A2  :  S  E2.  But  the  same  quantity  of  light,  being  diffused 
over  A  C  and  E  G,  will  be  more  intense,  as  the  surface  is  smaller. 
Hence,  the  intensity  of  light  at  E  :  intensity  at  A  : :  A  C  :  E  G  : : 
JS  A2  :  S  E2,  which  proves  the  proposition.  This  demonstration 
is  applicable  to  every  kind  of  emanation  in  straight  lines  from  a 
point. 

If  the  surface  receiving  the  light  be  oblique  to  the  axis  of  the 
beam,  the  intensity  of  illumination  is  proportional  to  the  sine  of 
the  angle  which  the  rays  make  with  the  surface.     Let  Fig.  208 
represent  a  section  through  the  axis 
of  a  beam  passing  through  an  orifice 
a  b  and  falling  upon  the  inclined 

surface  at  c  d.     Now  because  of  the     _^ a|_ 

obliquity,  the  surface  c  d  is  greater      j> 

than  the  section  at  right  angles  to 

the  beam  represented  by  e  c,  and 

hence  is  less  intensely  illuminated 

at  any  point.     But  surface  e  c  :  surface  c  d  : :  line  e  c  :  line  c  d  : : 

sine  e  d  c  :  sine  c  e  d. 

Calling  the  illumination  upon  any  point  of  the  right  section 
e  c  unity  or  u,  the  illumination  upon  any  point  of  c  d  will  be 

e  c 

u  x  —j  =  u  x  sine  e  d  c. 
cd 

344.  Brightness  the  Same  at  all  Distances.— The  bright- 


228  OPTICS. 

ness  of  an  object  is  the  quantity  of  light  which  it  sheds,  as  com- 
pared with  the  apparent  area  from  which  it  comes.  Now  the  quan- 
tity (or  intensity),  as  has  just  been  shown,  varies  inversely  as  the 
square  of  the  distance.  The  apparent  area  of  a  given  surface  also 
diminishes  in  the  same  ratio,  as  we  recede  from  it.  Hence  the 
brightness  is  constant.  For  illustration,  if  we  remove  to  three 
times  the  distance  from  a  luminous  body,  we  receive  into  the 
eye  nine  times  less  light,  but  the  body  also  appears  nine  times 
smaller,  so  that  the  relation  of  light  to  apparent  area  remains  the 
same. 

345.  Loss  of  Intensity  by  Absorption.—  In  a  uniform 
medium,  while  the  distance  increases  arithmetically,  the  intensity 
diminishes  geometrically.  Imagine  the  medium  to  be  divided  by 
parallel  planes  into  strata  of  equal  thickness  ;  and  suppose  the 

first  stratum  to  diminish  the  intensity  by  -  of  the  whole.  Then 
the  intensity  of  the  light  which  reaches  the  second  stratum  is 

1  --  =  -  .     But  on  account  of  the  uniformity  of  the  me- 
n          n 

dium,  every  stratum  produces  the  same  effect,  that  is,  it  transmits 

to  the  next,  -  of  that  which  falls  upon  it.    Therefore,  —  ^- 
n  n 

1        (n—  I)2    .  (n  —  I)3  ., 

-,  or  3  —  —  L.     leaves    the    second    stratum/  —     —  -,  the 
n2  ns 

third,  and  so  on,  in  a  geometrical  series.  For  example,  if  a  piece 
of  colored  glass  is  1J  inch  thick,  and  each  quarter  of  an  inch 
absorbs  f  of  the  light  which  falls  upon  it,  then  about  one-hun- 
dredth of  what  enters  the  first  surface  will  escape  from  the  last. 


For  ^-  )  =  .01  nearly. 

346.  Photometers.  —  These  are  instruments  designed  for  the 
measurement  of  the  relative  intensities  of  light.  We  cannot  de- 
termine by  the  eye  alone  how  many  times  more  intense  one  light 
is  than  another,  though  we  can  judge  with  tolerable  accuracy 
when  two  surfaces  are  equally  illuminated.  Photometers  are, 
therefore,  generally  constructed  on  the  plan  of  determining  the 
ratio  of  intensities  of  two  lights,  by  means  of  our  ability  to  decide 
when  they  illuminate  two  surfaces  equally.  It  is  sufficient  to 
mention  Rumford's  method.  Let  the  two  unequal  lights  be 
placed  at  A  and  B  (Fig.  209)  so  that  the  shadows  of  an  opaque 
rod  C  shall  fall  side  by  side  upon  a  screen  as  at  a  and  b.  The 
portion  of  the  screen  upon  which  the  shadow  a  falls  receives  light 
only  from  the  candle  B  and  none  from  the  gas  flame  A  ;  the  por- 


SHADOWS. 


229 


tion  I  is  illuminated  by  A  alone.    The  opaque  body  thus  secures 
for  each  light  a  portion  of  the  screen  which  it  alone  illuminates. 


FIG.  209. 


Now  move  either  light  towards  or  from  the  screen  until  the  two 
portions  a  and  b  are  equally  illuminated  by  their  respective  lights, 
and  then  measure  the  distances  from  A  to  b,  =  m,  and  from  B  to 
a,  =  n. 

B  at  distance  n  illuminates  the  screen  as  intensely  as  A  at  dis- 
tance m. 

Calling  B'  the  illumination  by  B  at  distance  n,  and  A'  that  of 
A  at  distance  m,  we  have  A  =  B'.  If  B  were  moved  back  to  dis- 
tance m,  then,  according  to  Art.  343,  its  power  of  illuminating,  or 

n2  n2 

intensity,  would  be  — ^  x  B  =  B    =—~A' ;  hence, 
m  m 

B"  :Ar::n*i  w2; 

or,  the  intensities  vary  directly  as  the  squares  of  those  distances 
from  the  screen  at  ivhich  equal  illumination  is  obtained. 

347.  Shadows. — When  a  luminous  body  shines  on  one  which 
is  opaque,  the  space  beyond  the  latter,  from  which  the  light  is 
excluded,  is  called  a  shadow.  The  same  word,  as  commonly  used, 
denotes  only  the  section  of  a  shadow  made  by  a  surface  which 
crosses  it.  Shadows  are  either  total  or  partial.  If  tangents  are 
drawn  on  all  the  corresponding  sides  of  the  two  bodies,  the  space 
inclosed  by  them  beyond  the  opaque  body  is  the  total  shadow  ;  if 
other  tangents  are  drawn,  crossing  each  other  between  the  bodies, 
the  space  between  the  total  shadow  and  the  latter  system  of  tan- 
gents is  the  partial  shadow,  or  penumbra.  In  case  the  bodies  are 
spheres,  as  in  Fig.  210,  the  total  shadow  will  be  a  cylinder,  or  con- 
ical frustum,  each  of  infinite  length,  or  a  complete  cone,  accord- 
ing to  the  relative  size  of  the  spheres.  But,  in  every  case,  the 
penumbra  and  inclosed  total  shadow  will  form  an  increasing  frus- 
tum. It  is  obvious  that  the  shade  of  the  penumbra  grows 


230 


OPTICS. 


gradually  deeper  from  the  outer  surface  to   the  total  shadow 
within  it. 

Every  shadow  cast  by  the  sun  has  a  penumbra  bordering  it, 
which  gives  to  the  shadow  an  ill-defined  edge ;  and  the  more 

Flo.  210. 


remote  the  sectional  shadow  is  from  the  opaque  body  which  casts 
it,  the  broader  will  be  the  partial  shadow  on  the  edge. 

If  instead  of  a  luminous  body  of  sensible  magnitude,  the 
source  of  light  be  a  point,  then  no  penumbra  will  be  formed. 
The  electric  arc  between  carbon  points  casts  a  sharply-defined 
shadow  of  a  hair  upon  a  screen  placed  at  a  great  distance. 


CHAPTER    II. 

REFLECTION  OF  LIGHT. 

348.  Radiant  and  Specular  Reflection.— Light  is  said  to 
be  reflected  when,  on  meeting  a  surface,  it  is  turned  back  into  the 
same  medium.  In  ordinary  cases  of  reflection,  the  light  is  diffused 
in  all  directions,  and  it  is  by  means  of  the  light  thus  scattered 
from  a  body  that  it  becomes  visible,  when  it  sheds  no  light  of  its 
own.  This  is  called  radiant  reflection.  It  is  produced  by  unpol- 
ished surfaces.  But  when  a  surface  is  highly  polished,  a  beam  of 
light  falling  on  it  is  reflected  in  some  particular  direction  ;  and, 
if  the  eye  is  placed  in  this  reflected  beam,  it  is  not  the  reflecting 
surface  which  is  seen,  but  the  original  object,  apparently  in  a  new 
position.  This  is  called  specular  reflection.  It  is,  however,  gener- 
ally accompanied  by  some  degree  of  radiant  reflection,  since  the 
reflector  itself  is  commonly  visible  in  all  directions.  Ordinary 


LAW    OF    REFLECTION. 


231 


mirrors  are  not  suitable  for  accurate  experiments  on  reflection, 
because  light  is  modified  by  the  glass  through  which  it  passes. 
The  speculum  is  therefore  used,  which  is  a  reflector  made  of  solid 
metal,  and  accurately  ground  to  any  required  form,  either  plane, 
convex,  or  concave.  The  word  mirror  is,  however,  much  used  in 
optics  for  every  kind  of  reflector. 

Optical  experiments  are  usually  performed  on  a  beam  of  light 
admitted  through  an  aperture  into  a  darkened  room ;  the  direc- 
tion of  the  beam  being  regulated  by  an  adjustable  mirror  placed 
outside.  An  instrument  consisting  of  a  plane  speculum  moved  by 
a  clock,  in  such  a  manner  that  the  reflected  sunbeam  shall  remain 
stationary  at  all  hours  of  the  day,  is  called  a  heliostat. 

349.  The  Law  of  Reflection. — When  a  ray  of  light  is  inci- 
dent on  a  mirror,  the  angle  between  it  and  a  perpendicular  to  the 
surface  at  the  point  of  incidence,  is  called  the  angle  of  incidence; 
and  the  angle  between  the  reflected  ray  and  the  same  perpendicu- 
lar, is  called  the  angle  of  reflection.  The  law  of  reflection  found 
to  be  universally  true  is  the  following: 

The  incident  ray,  the  reflected  ray,  and  the  normal  to  the  sur- 
face are  in  the  same  plane,  and  the  normal  bisects  the  angle  which 
these  rays  make  with  each  other. 

This  is  well  shown  by  attaching  a  small  mirror  to  the  centre 
of  a  graduated  semicircle  perpendicular  to  its  plane.  Let  M  D  N 
(Fig.  211)  be  the  semicircle,  graduated  from  D  both  ways  to  M 
and  N,  and  mounted  so  that  it 
can  be  revolved  on  its  centre,  and 
clamped  in  any  position.  Let  the 
small  mirror  be  at  C,  with  its  plane 
perpendicular  to  C  D  ;  then  a  ray 
from  the  heliostat,  as  A  C,  passing 
the  edge  at  a  particular  degree, 
will  be  seen  after  reflection  to  pass 
the  corresponding  degree  in  the 
other  quadrant.  By  revolving  the 
semicircle,  any  angle  of  incidence 
may  be  tried,  and  the  two  rays  are 
always  found  to  be  in  the  same 
plane  with  C  D,  and  equally  in- 
clined to  it.  % 

As  the  mirror  revolves,  the  re- 
flected ray  revolves  tivice  as  fast. 

For  A  C  D  is  increased  or  diminished  by  the  angle  through 
which  the  mirror  turns ;  therefore  D  C  B  is  also  increased  or 
diminished  by  the  same  ;  hence  A  C  B,  the  angle  between  the  two 


FIG.  211. 


232  OPTICS. 

rays,  is  increased  or  diminished  by  the  sum  of  both,  or  twice  the 
same  angle. 

It  follows  from  the  law  of  reflection,  that  a  ray  which  falls  on 
a  mirror  perpendicularly,  retraces  its  own  path  after  reflection.  It 
is  obvious,  also,  that  the  complements  of  the  angles  of  incidence 
and  reflection  are  equal,  i.  e.  A  C  M  —  B  0  N.  The  law  of  reflec- 
tion is  applicable  to  curved  as  well  as  to  plane  mirrors  ;  the  radius 
of  curvature  at  any  point  being  the  perpendicular  with  which  the 
incident  and  reflected  rays  make  equal  angles. 

Radiant  reflection  forms  no  exception  to  the  foregoing  law, 
though  the  incident  rays  are  in  one  and  the  same  direction,  and 
the  reflected  rays  are  scattered  every  way.  For  the  minute  cavi- 
ties and  prominences  which  constitute  the  roughness  of  the  gen- 
eral surf  ace  are  bounded  by  small  surfaces  lying  at  all  inclinations; 
and  each  one  reflecting  the  rays  which  meet  it  in  accordance  with 
the  law,  those  rays  are  necessarily  thrown  off  in  all  possible  direc- 
tions. 

The  proportion  of  the  incident  light  reflected  varies  with 
the  angle  of  incidence.  When  light  strikes  the  surface  of  water 
perpendicularly  only  .018  is  reflected,  the  rest  entering  the  water  ; 
but  at  an  incidence  of  89 £°  . 721  is  reflected.  In  the  case  of  mer- 
cury .666  is  reflected  at  perpendicular  incidence,  while  .721  is 
reflected  at  89-|°,  the  non-reflected  rays  entering  the  metal  and 
being  destroyed,  as  light. 

350.  Inclination  of  Rays  to  each  other  not  altered  by 
the  Plane  Mirror. — 

1.  Rays  which  diverge  before  reflection,  diverge  at  the  same 
angle  after  reflection. 

Let  M  N  (Fig.  212)  be  a  plane  mirror,  and  A  B,  A  C,  any  two 
rays  of  light  falling  upon  it  from  the 
radiant  A,  and  reflected  in  the  lines 
BE,  C  G.  Draw  the  perpendicular 
A  P,  and  produce  it  indefinitely,  as  to  F, 
behind  the  mirror ;  also  produce  the  re- 
flected rays  back  of  the  mirror.  Let  Q  R 
be  perpendicular  to  the  mirror  at  the 
point  B  ;  it  is  therefore  parallel  to  A  F, 
and  the  plane  passing  through  A  F  and 
Q  R,  is  that  which  includes  the  ray 
A  B,  B  E.  Therefore  E  B,  when  pro- 
duced back  of  the  mirror,  intersects  A  P 
produced.  Let  J^be  the  point  of  intersection.  B  A  F  =  A  B  Q, 
and  A  FB  =  EB  Q-,  but  A  B  Q  =  EB  Q  (Art.  349)  ;  .-.  BA  F 
=  A  FB,  and  A  B  =  F  B.  If  P  and  B  be  joined,  P  B  being  in 


SPHERICAL    MIRRORS.  333 

the  plane  M  N  is  perpendicular  to  A  F,  and  therefore  bisects  it. 
Hence,  the  reflected  ray  meets  the  perpendicular  A  Fas  far  behind 
the  mirror,  as  the  incident  ray  does  in  front.  In  the  same  way  it 
may  be  proved  that  A  C  =  C  F,  and  that  C  G,  when  produced 
back  of  the  mirror,  meets  A  Fat  the  same  point  F. 

Now,  since  the  triangles  A  C  B  and  F  C  B,  have  their  sides 
respectively  equal,  their  angles  are  equal  also  ;  hence  B  A  C  = 
B  F  C.  Therefore  any  two  rays  diverge  at  the  same  angle  after 
reflection  as  they  did  before  reflection. 

Since  the  reflected  rays  seem  to  emanate  from  F,  that  point  is 
called  the  apparent  radiant ;  A  is  the  real  radiant. 

2.  Rays  which  converge  before  reflection,  converge  at  the  same 
angle  after  reflection.     Let  E  B,  G  C,  be  incident  rays  converging 
toward  F,  and  let  B  A,  C  A,  be  the  reflected  rays.     It  may  be 
proved  as  before,  that  A  and  F  are  in  the  same  perpendicular, 
A  F,  and  equidistant  from  P,  and  that  E  F  G  =  B  A  C. 

The  point  F,  to  which  the  incident  rays  were  converging,  is 
called  the  virtual  focus  ;  A  is  the  real  focus. 

3.  Rays  which  are  parallel  before  reflection  are  parallel  after 
reflection. 

It  has  been  proved  in  case  1,  that  F,  the  intersection  of  the 
reflected  rays,  is  as  far  behind  the  mirror,  as  A,  the  intersection 
of  incident  rays,  is  before  it.  Now,  if  the  incident  rays  are  parallel, 
A  is  at  an  infinite  distance  from  the  mirror.  Therefore  F  is  at 
an  infinite  distance  behind  ifc,  and  the  reflected  rays  are  parallel. 

In  all  cases,  therefore,  rays  reflected  by  a  plane  mirror  retain 
the  same  inclination  to  each  other  which  they  had  before  reflec- 
tion. 

351.  Spherical  Mirrors. — A  spherical  mirror  is  one  which 
forms  a  part  of  the  surface  of  a  sphere,  and  is  either  convex  or 
concave.     The  axis  of  such  a  mirror  is  that  radius  of  the  sphere 
which  passes  through  the  middle  of  the  mirror.     In  the  practical 
use  of  spherical  mirrors,  it  is  found  that  the  light  must  strike  the 
surface  very  nearly  at  right  angles ;  hence,  in  the  following  state- 
ments, the  mirror  is  supposed  to  be  a  very  small  part  of  the  whole 
spherical  surface,  and  the  rays  nearly  coincident  with  the  axis. 

It  is  sufficient  to  trace  the  course  of  the  rays  on  one  side  of  the 
axis,  since,  on  account  of  the  symmetry  of  the  mirror  around  the 
axis,  the  same  effect  is  produced  on  every  side. 

352.  Converging  Effect  of  a  Concave  Mirror.— 

1.  Parallel  rays  are  converged  to  the  middle  point  between  the 
centre  and  surface,  which  is  therefore  called  the  focus  of  parallel 
rays  or  the  principal  focus.  Let  R  A,  LE  (Fig.  213),  be  parallel 


234 


OPTICS. 


rays  incident  upon  the  concave  mirror  A  B,  whose  centre  of  con- 
cavity is  C.     The  ray  L  E,  passing  through   0,  and  therefore 


FIG.  213. 


perpendicular  to  the  mirror  at  E,  is  reflected  directly  back.  Join 
C  A,  and  make  C  A  F  =  R  A  (7;  then  R  A  is  reflected  in  the 
line  A  F,  and  the  two  reflected  rays  meet  at  F.  R  A  C  =  A  C  F, 
.'.  A  C  F  =  FA  C,  and  A  F  —  OF,  and  as  A  and  j^are  very 
near  together,  E  F '=  A  F  =  F  C\  that  is,  the  focus  of  parallel 
rays  is  at  the  middle  point  between  C  and  E. 

2.  Diverging  rays,  falling  on  a  given  concave  mirror,  are  re- 
flected converging,  parallel,  or  less  diverging,  according  to  the 
degree  of  divergency  in  the  original  pencil.  Let  C  (Fig.  214)  be 
the  centre  of  concavity,  and  F  the  focus  of  parallel  rays.  Then, 

FIG.  214. 


rays  diverging  from  any  point,  A,  beyond  (7,  will  be  converged  to 
some  point,  a,  between  C  and  F,  since  the  angles  of  incidence  and 
reflection  are  less  than  those  for  parallel  rays.  Rays  diverging 
from  0  are  reflected  back  to  (7;  those  from  points  between  C  and 
F,  as  #,  are  converged  to  points  beyond  C,  as  A  ;  those  diverging 
from  F  become  parallel ;  and  those  from  points  between  F  and  the 
mirror,  as  D,  diverge  after  reflection,  but  at  a  less  angle  than  be- 
fore, and  seem  to  flow  from  A'.  To  prove,  in  the  last  case,  that 
the  angle  of  divergence,  A',  after  reflection,  is  less  than  the  angle 
Z>,  the  divergence  before  reflection,  observe  that  the  angle  A'  is 
less  than  the  exterior  angle  H  B  C,  or  its  equal,  DBG  (Art. 
349)  ;  and  D  B  (7 is  less  than  the  exterior,  A'  D  B\  much  more, 
then,  is  A'  less  than  A'  D  B. 

3.  Converging  rays  are  made  to  converge  more.  The  rays  H  B, 
A  E,  converging  to  A ',  are  reflected  to  D,  nearer  the  mirror  than 
Fis.  And  it  has  been  shown  that  the  angle  D  is  larger  than  A', 
hence  the  convergency  is  increased. 

From  the  three  foregoing  cases,  it  appears  that  the  concave 


CONJUttATE    FOCI. 


235 


mirror  always  tends  to  produce  convergence/  ;  since,  when  it  does 
not  actually  produce  it,  it  diminishes  divergency. 

The  principal  focus  can  be  determined  practically  by  receiving 
the  sun's  rays  upon  the  mirror,  parallel  to  its  axis,  and  finding 
the  point  at  which  a  sharp  image  of  the  sun  is  formed.  The  dis- 
tance of  this  image  from  the  surface  is  one-half  the  radius  of  cur- 
vature. 

353.  Conjugate  Foci. — When  light  radiates  from  A,  it  is 
reflected  to  a  ;  when  it  radiates  from  a,  it  meets  at  A.  Any  two 
such  interchangeable  points  are  called  conjugate  foci.  If  the  radius 
of  the  mirror  and  the  distance  of  one  focus  from  the  mirror  are 
given,  the  distance  of  its  conjugate  focus  may  be  determined. 
Let  the  radius  =  r ;  the  distance  A  E  •=,  m ;  and  a  E  =  n.  As 
the  angle  A  B  a  is  bisected  by  B  C,  A  B  :  a  B  : :  A  C  :  a  C ;  that 
is,  since  B  E  is  very  small,  A  E  :  a  E : :  A  C  i  a  C,  or,  mini: 
m  —  r  :  r  —  n. 


m  =  — • 


n  r 


;  and  n  =  -- 


m  r 


FIG.  215. 


2  n  -  r  '  -  2  m  -  r 

If  A  is  not  on  the  axis  of  the  mirror,  as  in  Fig.  215,  let  a  line 
be  drawn  through  A  and  (7,  meet- 
ing the  mirror  in  E ;  this  is  called 
a  secondary  axis,  and  the  light 
radiating  from  A  will  be  reflected 
to  a  on  the  same  secondary  axis, 
for  A  E  is  perpendicular  to  the 

mirror,  and  will  be  reflected  directly  back  ;  and  if  A  E  and  G  E 
are  given,  a  E  may  be  found  as  before. 

354.  Diverging  Effect  of  a  Convex  Mirror. — 

1.  Parallel  rays  are  reflected  diverging  from  the  middle  point 
between  the  centre  and  surface.  Let  C(F\g.  216)  be  the  centre 
of  convexity  of  the  mirror  M  N,  and  draw  the  radii,  C  M,  CD, 

FIG.  216. 


producing  them  in  front  of  the  mirror  ;  these  are  perpendicular  to 
the  surface.  The  ray  R  D  will  be  reflected  back ;  A  M  will  be 
reflected  in  M  B,  making  B  M  E  =  A  M  E.  Produce  the  re- 
flected ray  back  of  the  mirror,  and  it  will  meet  the  axis  in  F,  mid- 
way from  <7toZ>;  for  FCM  =A  ME,  and  F  M  C=  B  M  E; 
therefore  the  triangle  F  C  Mis  isosceles,  and  C F '=  F '  M,  and  as 


236  OPTICS. 

M  is  very  near  7),  C  F  =  F  D.  Hence  the  rays,  after  reflection, 
diverge  as  if  they  radiated  from  a  point  in  the  middle  of  C  D, 
which  is  the  apparent  radiant. 

2.  Diverging  rays  have  their  divergency  increased.     Let  A  D, 
A  M  (Fig.  217),  be  the  diverging  rays  ;  D  A,  M  B,  the  reflected 

FIG.  217. 


rays ;  these  when  produced  meet  at  F,  which  is  the  apparent  radi- 
ant. MAFia  the  divergency  of  the  incident  rays,  and  A  F  B 
of  the  reflected  rays.  Now  the  exterior  angle,  A  F  B,  is  greater 
than  G  M  F,  or  B  M  E,  or  A  M  E.  But  A  M  E,  being  exterior,  is 
greater  than  M  A  F\  much  more,  then,  isAFU  greater  than  M  A  F. 
3.  Convergent  rays  are  at  least  rendered  less  convergent,  and 
may  become  parallel  or  divergent,  according  to  the  degree  of  pre- 
vious convergency.  The  two  first  effects  are  shown  by  Figs.  216 
and  217,  reversing  the  order  of  the  rays.  And  it  is  easy  to  per- 
ceive that  rays  converging  to  (7,  will  diverge  from  C  after  reflec- 
tion ;  if  to  a  point  more  distant  than  C,  they  will  diverge  after- 
ward from  a  point  between  (7  and  F  (Fig.  216),  and  vice  versa. 

The  general  effect,  therefore,  of  a  convex  mirror,  is  to  produce 
divergency. 

A  and  F  (Fig.  217)  are  called  conjugate  foci,  being  inter- 
changeable points ;  for  rays  from  A  move  after  reflection  as  though 
from  F,  and  rays  converging  to  F  are  by  reflection  converged  to 
A.  Conjugate  foci,  in  the  case  of  the  convex  mirror,  are  in  the 
same  axis  either  principal  or  secondary,  as  they  are  in  the  concave 

mirror,  and  for  the  same  reason, 
IG<  viz.,  that  every  axis  is  perpendic- 

ular to  the  surface. 

Their  relative  positions  may 
be  determined  by  the  formula, 

~A* —  ~^L^  easily  deduced,  as  in  Art.  353, 

n  r 


*C 


To  determine  the  radius  of  cur- 
vature experimentally:  Through 
a  circular  opening  in  a  screen 
whose  diameter  is  greater  than 
E  H  (Fig.  218),  receive  the  sun's 
rays  upon  the  mirror,  parallel  to  the  axis,  and  move  the  screen  so 


Y* 


IMAGES    BY    A    PLANE    MIRROR.  237 

that  the  diameter  K D  of  the  illuminated  circle  is  twice  the  chord 
E  H  of  the  mirror ;  then  measure 

D  ff  =  F H=  PL  =  i  Kad. 

To  render  this  method  more  accurate  cover  all  of  the  mirror, 
except  a  small  central  circle,  with  some  opaque  covering,  and  use 
only  the  exposed  portion  as  above. 

355.  Images  by  Reflection. — An  optical  image  consists  of 
a  collection  of  focal  points,  from  which  light  either  really  or  appa- 
rently radiates.     When  rays  are  converged  to  a  focus  they  do  not 
stop,  but  cross,  and  diverge  again,  as  if  originally  emanating  from 
the  focal  point.    A  collection  of  such  points,  arranged  in  order, 
constitutes  a  real  image.     When  rays  are  reflected  diverging,  they 
proceed  as  though  they  emanated  from  a  point  behind  the  mirror. 
A  collection  of  such  imaginary  radiants  forms  an  apparent  or 
virtual  image.   The  images  formed  by  plane  and  convex  mirrors  are 
always  apparent ;  those  formed  by  concave  mirrors  may  be  of  either 
kind. 

356.  Images  by  a  Plane  Mirror. — When  an  object  is  before 
a  plane  mirror,  its  image  is  at  the  same  distance  behind  it,  of  the 
same  magnitude,  and  equally  inclined  to  it.     Let  M  N  (Fig.  219) 
be  a  plane  mirror,  and  A  B  an  ob- 
ject before  it,  and  let  the  position  FIG.  219. 

of  the  object  be  such  that  the  re- 
flected rays  may  enter  the  eye 
placed  at  H.  From  A  and  B  let 
fall  upon  the  plane  of  the  mirror 
the  perpendiculars  A  E,  B  G,  and 
produce  them,  making  E  a  =  A  E, 
and  Q  b  =  B  G.  Now,  since  the 
rays  from  A  will,  after  reflection, 

radiate  as  if  from  a  (Art.  350),  and  those  from  B,  as  if  from  I,  and 
the  same  of  all  other  points,  therefore  the  image  and  object  are 
equally  distant  from  the  mirror.  A  C,  a  c,  parallel  to  the  mirror, 
are  equal ;  as  B  G  —  b  G,  and  A  E  =  a  E,  therefore,  by  sub- 
traction, E  C  —  b  c ;  also  the  right  angles  at  C  and  c  are  equal. 
Therefore  A  B  —  a  b,  and  B  A  C=b  ac',  that  is,  the  object  and 
image  are  of  equal  size,  and  equally  inclined  to  the  mirror. 

It  appears  from  the  demonstration,  that  the  object  and  its 
image  are  comprehended  between  the  same  perpendiculars  to  the 
plane  of  the  mirror;  and  this  image  will  appear  in  the  same 
position  whatever- may  be  the  position  of  the  eye. 

The  object  and  image  obviously  have  to  each  other  twice  the 
inclination  that  each  has  to  the  mirror.  Hence,  in  a  mirror  in- 
clined 45°  to  the  horizon,  a  horizontal  surface  appears  vertical, 
and  one  which  is  vertical  appears  horizontal. 


238 


OPTICS. 


FIG.  220. 


357.  Symmetry  of  Object  and  Image. — All  the  three  di- 
mensions .of  the  object  and  image  are  respectively  equal,  as  shown 
above,  but  one  of  them  is  inverted  in  position,  namely,  that  dimen- 
sion which  is  perpendicular  to  the  mirror.     Hence,  a  person  and 
his  image  face  in  opposite  directions ;  and  trees  seen  in  a  lake  have 
their  tops  downward.     Those  dimensions  which  are  parallel  to  the 
mirror  are  not  inverted.     In  consequence  of  the  inversion  of  one 
dimension  alone,  the  object  and  its  image  are  not  similar,  but 
symmetrical  forms ;  and  one  could  not  coincide  with  the  other  if 
brought  to  occupy  the  same  space.    The  image  of  a  right  hand  is 
a  left  hand,  and  all  relations  of  right  and  left  are  reversed.     It  is 
for  this  reason  that  a  printed  page,  seen  in  a  mirror,  is  like  the 
type  with  which  it  was  printed. 

358.  The  Length  of  Mirror  Requisite  for  Seeing  an 
Object. — If  an  object  is  parallel  to  a  mirror,  the  length  of  mirror 
occupied  by  the  image  is  to  the  length  of  the  object  as  the  reflected 

ray  to  the  sum  of  the  incident  and  reflected 
rays.  Let  A  B  (Fig.  220)  be  the  length  of  the 
object,  CD  that  of  the  image,  and  FG  that  of 
the  space  occupied  on  the  mirror ;  then,  by 
similar  triangles,  FG  :  CD::  EF:  EC.  But 
CD  =  AB,  and  CF=  A  F'y .%  FG  :  AB:: 
E  F  :  A  F  -f-  F  E.  If  the  eye  is  brought 
nearer  the  mirror,  the  space  on  the  mirror 
occupied  by  the  image  is  diminished,  because 

77 ^ has  to  A  F+  F  E  a  less  ratio  than  before.    The  same  effect  is 

produced  by  removing  the  object 

further  from   the  mirror.     The 

length  of  mirror  necessary  for  a 

person  to  see  himself  is  equal  to 

half  his  height,  because  in  that 

case,  E  F:A  F  +  FU::l  :  2, 

which  ratio  will  not  be  altered 

by  change  of  distance. 

359.  Displacement  of  Im- 
age by  Two  Reflections. — If 

an  image  is  seen  by  light  reflected 
from  two  mirrors  in  a  plane  per- 
pendicular to  their  common  sec- 
tion, its  angular  deviation  from 
the  object  is  equal  to  twice  the 
inclination  of  the  mirrors.  Let 
A  By  CD  (Fig.  221)  be  two  plane 


FIG.  221. 


MULTIPLIED     IMAGES    BY    TWO    MIRRORS.       239 

mirrors  inclined  at  the  angle  A  G  C.  If  an  eye  at  H  sees  the  star 
8  in  the  direction  0,  the  angle  SHO  =  2AGC. 

For  the  exterior  angle  C  D  B  —  I  =  a  +  G,  or  2  #  =  2  a  -f- 
2  G,  and  B  D  0=  2b  =  2a+  H;  hence  2a  +  H=2a  +  2  £ ; 
therefore  H  =  2  G. 

This  principle  is  employed  in  the  construction  of  Hartley's 
quadrant,  and  the  sextant,  used  at  sea  for  measuring  angular  dis- 
tances. The  angles  measured  are  twice  as  great  as  the  arc  passed 
over  by  the  index  which  carries  the  revolving  mirror ;  hence,  in 
the  quadrant,  an  arc  of  45°  is  graduated  into  90° ;  and,  in  the  sex- 
tant, an  arc  of  60°  is  graduated  into  120°. 

360.  Multiplied  Images  by  Two  Mirrors. — 

1.  Parallel  Mirrors.  The  series  of  images  is  infinite  in  num- 
ber, and  arranged  in  a  straight  line,  perpendicular  to  the  mirrors. 
The  object  a,  between  the  parallel  mirrors,  A  and  B  (Fig.  222), 
has  an  image  at  a',  as  far  behind  A  as  a  is  in  front  of  it.  To 

FIG.  222. 


I 


avoid  confusion,  a  pencil  from  only  one  point  o  is  drawn,  once  re- 
flected at  c,  and  entering  the  eye  as  though  it  came  from  o'.  The 
rays  reflected  by  A  diverge  as  though  they  emanated  from  a'\ 
hence,  the  light  reflected  from  A  upon  B  may  be  regarded  as  pro- 
ceeding from  a  real  object  at  a',  whose  image  will  be  b,  as  far  back 
of  B  as  a'  is  in  front  of  B.  The  light  reflected  from  B  to  A  again 
diverges  as  though  it  really  came  from  b,  and  regarding  b  as  a 
real  object  as  before  Us  image  would  be  formed  at  a"  as  far  behind 
A  as  b  is  in  front  of  it.  The  pencil  which  enters  the  eye  seems 
to  proceed  from  o'",  having  been  reflected  from  e",  as  though  it 
ca:n9  from  o",  its  reflection  in  this  case  having  been  from  e'  as 
though  it  came  from  o',  though  it  was  really  reflected  from  e  after 
having  emanated  from  o.  The  pencil  which  would  enter  the  eye 
from  a  third  image  at  the  left  of  a"  may  be  traced  through  all  its 
reflections  in  like  manner.  As  light  is  absorbed  and  scattered  at 
•each  reflection  the  number  of  such  images  is  limited. 

The  multiplied  images  of  a  small  bright  object,  sometimes 


240 


OPTICS. 


seen  in  a  looking-glass,  are  produced  by  repeated  reflections  be- 
tween the  front  and  the  silvered  covering  on  the  back  side.  At 
each  internal  impact  on  the  first  surface  some  light  escapes,  and 
shows  us  an  image,  while  another  portion  is  reflected  to  the  back, 
and  thence  forward  again.  The  image  of  a  lamp  viewed  very 
obliquely  in  a  mirror  is  sometimes  repeated  eight  or  ten  times  ;  and 
a  planet,  or  bright  star,  when  seen  in  a  looking-glass,  will  be  ac- 
companied by  three  or  four  faint  images,  caused  in  the  same  way. 
2.  Inclined  Mirrors.  Let  Q  (Fig.  223)  be  the  object,  and  0 
the  position  of  the  eye.  Wit]^  R  as  a  centre  and  radius  R  Q,  de- 
scribe a  circumference. 

Suppose  a  chord  Q  A  to  be  drawn  perpendicular  to  the  mirror 

8,    then   A   will    be    the 

FIG.  223.  image   of    Q.     Regarding 

JL>  A  as  a  real  object,  as  in 

the  case  of  parallel  mirrors, 
draw  a  chord  A  B  per- 
pendicular to  the  mirror 
T,  then  since  o  B  =  o  A, 
B  will  be  the  image  of  A. 
Suppose  a  chord  B  (7  to  be 
drawn  perpendicular  to  the 
mirror  JS,  then  (7,  being 
as  far  behind  the  mir- 
ror S  as  the  object  B, 
assumed  as  real,  is  in  front 
of  it,  will  be  the  image  of 
B ;  and  for  like  reasons 
D  will  be  the  image  of  0 
in  mirror  T.  All  the 
images  formed  by  the  inclined  mirrors  are  thus  seen  to  be  confined 
to  the  circumference  of  a  circle  described  as  above  stated.  There 
can  be  no  image  of  D,  since  it  lies  behind  both  mirrors  prolonged. 
The  image  A  is  seen  by  rays  which  proceed  from  Q  to  a  and  thence 
to  the  eye  at  0.  B  is  seen  as  though  the  rays  came  from  B  to  0, 
these  having  been  reflected  at  c  as  though  they  came  from  A,  the 
reflection  at  b  being  direct  from  Q.  The  image  C  is  seen  by  rays 
reflected  from  the  points  /,  e,  d\  and  D  by  rays  reflected  from 
&,  if  h,  g.  The  reflections  occur  in  the  order  d,  e,f,  and  g,  h,  i,  Tc. 
Only  images  formed  by  light  first  reflected  from  S  have  been 
considered ;  a  second  series  produced  by  light  first  reflected  from 
T  may  be  constructed  in  like  manner. 

361.  The  Kaleidoscope. — This  instrument,  when  carefully 
constructed,  beautifully  exhibits  the  phenomenon  of  multiplied 


IMAGES    BY    THE    CONCAVE    MIRROR.          241 

reflection  by  inclined  mirrors.  It  consists  of  a  tube  containing 
two  long,  narrow,  metallic  mirrors,  inclined  at  a  suitable  angle  ry 
and  is  used  by  placing  the  objects  (fragments  of  colored  glassy 
&c.)  at  one  end,  and  applying  the  eye  to  the  other.  In  order  that 
there  may  be  perfect  symmetry  in  the  figure  made  up  of  the  ob- 
jects and  their  successive  images,  the  angle  of  the  mirrors  should 
be  of  such  size,  that  it  can  be  exactly  contained  an  even  number 
of  times  in  360°.  The  best  inclination  is  30°  ;  and  the  field  of 
view  is  then  composed  of  12  sectors.  It  is  also  essential,  that  the 
small  objects  forming  the  picture,  should  lie  at  the  least  possible 
distance  beyond  the  mirrors.  To  insert  three  mirrors  instead  of 
two,  as  is  often  done,  only  serves  to  confuse  the  picture,  and  mar 
its  beauty. 

362.  Images  by  the  Concave  Mirror.  —  The  concave 
mirror  forms  various  images,  either  real  or  apparent,  either 
greater  or  less  than  the  object,  either  erect  or  inverted,  according 
to  the  place  of  the  object. 

1.  The  object  between  the  mirror  and  its  principal  focus.  By 
Art.  352  (2),  rays  which  diverge  from  a  point  between  the  mirror 
and  its  principal  focus,  continue  to  diverge  after  reflection,  but  in 
a  less  degree.  Let  C  be  the  centre,  and  F  the  principal  focus  of 
the  mirror  M  N  (Fig.  224),  and  A  B  the  object.  Draw  the  axes, 
C  A,  C  B,  and  produce  them  behind  the  mirror.  The  pencil  from 


FIG. 


A  will  be  reflected  to  the  eye  at  H,  radiating  as  from  a,  in  the 
same  axis ;  likewise,  that  from  B,  as  from  b.  Therefore,  the 
image  is  apparent,  since  rays  do  not  actually  flow  from  it;  erect, 
as  the  axes  do  not  cross  each  other  between  the  object  and  image  ; 
enlarged,  because  it  subtends  the  angle  of  the  axes  at  a  greater  dis- 
tance than  the  object  does.  As  the  object  approaches,  and  finally 
reaches  the  principal  focus,  the  reflected  rays  approach  parallelism, 
and  the  image  departs  from  the  mirror,  till  it  is  at  an  infinite  dis- 
tance. 

Other  rays  than  those  given  in  the  figure  fall  upon  the  mirror 
from  A,  but  are  reflected  either  above  or  below  the  eye,  and  there- 
fore have  no  part  in  the  production  of  the  image,  and  for  that 
reason  are  omitted.  The  same  is  true  of  rays  from  every  other 
point  of  the  object. 
16 


242  OPTICS. 

2.  Object  between  the  principal  focus  and  the  centre.  As  soon 
as  the  object  passes  the  principal  focus,  the  rays  of  each  pencil  be- 
gin to  converge  ;  and  each  radiant  of  the  object  has  its  conjugate 
focus  in  the  same  axis  beyond  the  centre  (Art.  353). 

For  example  the  rays  diverging  from  the  point  A,  represented 
in  Fig.  225  by  full  lines,  after  reflection  are  converged  to  a  situ- 
ated somewhere  on  the  secondary  axis  A  C  a,  and  rays  from  B, 

FIG.  225. 


given  as  dotted  lines,  converge  finally  to  b  on  the  axis  B  Ob. 
The  images  of  intermediate  points  are  formed  in  the  same  way. 

If  an  observer  is  beyond  a  b,  the  rays,  after  crossing  at  the 
image,  will  reach  him,  as  though  they  originated  in  a  b  ;  or  if  a 
screen  is  placed  at  a  b,  the  light  which  is  collected  in  the  focal 
points  will  be  thrown  in  all  directions  by  radiant  reflection  from 
the  screen.  Hence,  the  image  is  real;  it  is  also  inverted,  because 
the  axes  cross  betw?,en  the  conjugate  foci ;  and  it  is  enlarged,  since 
it  subtends  the  angle  of  the  axes  at  a  greater  distance  than  the 
object  does.  That  b  C  is  greater  than  B  C,  is  proved  by  joining 
C  G,  which  bisects  the  angle  B  G  b,  and  therefore  divides  B  b  so 
that  B  C  :  C  b  : :  B  G  :  G  b.  As  G  b  is  greater  than  B  G,  so  C  b 
is  greater  than  B  C.  When  the  object  reaches  the  centre,  the 
image  is  there  also,  but  inverted  in  position,  since  rays  which  pro- 
ceed from  one  side  of  C,  are  reflected  to  the  other  side  of  it. 

3.  Object  beyond  the  centre.  This  is  the  reverse  of  (2),  the 
conjugate  foci  having  changed  places ;  a  b,  therefore,  being  the 
object,  A  B  is  its  image,'  real,  inverted,  diminished.  As  the  ob- 
ject removes  to  infinity,  the  image  proceeds  only  to  the  principal 
focus  F. 

363.  Illustrated  by  Experiment. — These  cases  are  shown 
experimentally  by  placing  a  lamp  close  to  the  mirror,  and  then 
carrying  it  along  the  axis  to  a  considerable  distance  away.  While 
the  lamp  moves  from  the  mirror  to  the  principal  focus,  its  image 
behind  the  mirror  recedes  from  its  surface  to  infinity;  we  may 
then  regard  it  as  being  either  at  an  infinite  distance  behind,  or  an 
infinite  distance  in  front,  since  the  rays  of  every  pencil  are  par- 


IMAGES     BY     THE     CONVEX     MIRROR.  343 

allel.  After  the  lamp  passes  the  principal  focus,  the  image  ap- 
pears in  the  air  at  a  great  distance  in  front,  and  of  great  size,  and 
they  both  reach  the  centre  together,  where  they  pass  each  other; 
and,  as  the  lamp  is  carried  to  great  distances,  the  image,  growing 
less  and  less,  approaches  the  principal  focus,  and  is  there  reduced 
to  its  smallest  size.  The  only  part  of  the  infinite  line  of  the  axis 
before  and  behind,  in  which  no  image  can  appear,  is  the  small 
distance  between  the  mirror  and  its  principal  focus. 

If  a  person  looks  at  himself,  so  long  as  he  is  between  the  mir- 
ror and  the  principal  focus,  he  sees  his  image  behind  the  mirror 
and  enlarged.  But  when  he  is  between  the  principal  focus  and 
centre,  the  image  is  real,  and  behind  him  ;  the  converging  rays  of 
the  pencils,  however,  enter  his  eyes,  and  give  r/i  indistinct  view 
of  his  image  as  if  at  the  mirror.  When  he  reaches  the  centre,  the 
pupil  of  the  eye  is  seen  covering  the  entire  mirror,  because  rays 
from  the  centre  are  perpendicular,  and  return  to  it  from  all  parts 
of  the  surface.  Beyond  the  centre,  he  sees  the  real  image  in  the 
air  before  him,  distinct  and  inverted. 

364.  Images  by  the  Convex  Mirror. — The  convex  mir- 
ror affords  no  variety  of  cases,  because  diverging  rays,  which  fall 
upon  it,  are  made  to  diverge  still  more  by  reflection.  In  Fig.  226 
the  pencil  from  A  is  reflected,  as  if  radiating  from  a  in  the  same 

FIG.  226. 


axis  A  C,  and  that  from  B,  as  from  b  in  the  axis  B  C ;  and  these 
apparent  radiants  are  always  nearer  the  surface  than  the  middle 
point  between  it  and  G  (Art.  354).  The  image  is  therefore 
apparent ;  it  is  erect',  since  the  axes  do  not  cross  between  the 
object  and  image ;  and  it  is  diminished,  as  it  subtends  the  angle 
of  the  axes  at  a  less  distance  than  the  object. 

As  in  Fig.  224,  rays  from  A  and  B,  which  after  reflection  pass 
above  or  below  the  eye,  have  been  omitted ;  only  that  portion  of 
the  mirror  from  which  rays  are  represented  as  being  reflected  has 
any  part  in  the  formation  of  the  image.  For  eyes  in  other  posi- 
tions other  rays  would  be  used,  still  seeming  to  come  from  the 
same  image  a  &. 


244  OPTICS. 

365.  Caustics  by  Reflection. — These  are  luminouo  curved 
surfaces,  formed  by  the  intersections  of  rays  reflected  from  a  hemi- 
spherical concave  mirror.     The  name  caustic  is  given  from  the 
circumstance  that  heat,  as  well  as  light,  is  concentrated  in  the 

focal  points  which  compose  it.  BAD 
(Fig.  227),  represents  a  section  of  the 
mirror,  and  B  F  D  of  the  caustic ;  the 
point  F,  where  all  the  sections  of  the 
caustic  through  the  axis  meet  each 
other,  is  called  the  cusp.  When  the 
incident  rays  are  parallel,  as  in  the 
figure,  the  cusp  is  at  the  principal 
focus,  that  is,  the  middle  point  be- 
tween A  and  C.  The  rays  near  the 
axis  R  A,  after  reflection  meet  at  the 
cusp  (Art.  352) ;  but  those  a  little  more 

distant  cross  them,  and  meet  the  axis  a  little  further  toward  A. 
And  the  more  distant  the  incident  ray  from  the  axis,  the  further 
from  the  centre  does  the  reflected  ray  meet  the  axis.  Thus  each 
ray  intersects  all  the  previous  ones,  and  this  series  of  intersection- 
constitutes  the  curve,  B  F.  The  curve  is  luminous,  because  it 
consists  of  the  foci  of  the  successive  pencils  reflected  from  the 
arc  A  B. 

If  the  incident  rays,  instead  of  being  parallel,  diverge  from  a 
lamp  near  by,  the  form  of  the  caustic  is  a  little  altered,  and  the 
cusp  is  nearer  the  centre.  This  case  may  be  seen  on  the  surface 
of  milk,  the  light  of  the  lamp  being  reflected  by  the  edge  of  the 
bowl  which  contains  it. 

If  parallel  or  divergent  light  falls  on  a  convex  hemispherical 
mirror,  there  will  be  apparent  caustics  behind  the  mirror ;  that  is, 
the  light  will  be  reflected  as  if  it  radiated  from  points  arranged 
in  such  curves. 

366.  Spherical  Aberration  of  Mirrors. — It  has  already 
been  mentioned  (Art.  351),  that  the  statements  in  this  chapter 
relating  to  focal  points  and  images,  as  produced  by  spherical  mir- 
rors, are  true  only  when  the  mirror  is  a  very  small  part  of  the 
whole  spherical  surface.    In  Art.  365  we  have  seen  the  effect  of 
using  a  large  part  of  the  spherical  surface — viz.,  the  rays  neither 
converge  to,  nor  diverge  from  a  single  point,  but  a  series  of  points 
arranged  in  a  curve.     This  general  effect  is  called  the  spherical 
aberration  of  a  mirror ;  since  the  deviation  of  the  rays  is  due  to 
the  spherical  curvature.     The  deviation,  as  we  have  seen,  is  quite 
apparent  in  a  hemisphere,  or  any  considerable  portion  of  one  ;  but 
it  exists  in  some  degree  in  any  spherical  mirror,  unless  infinitely 
small  compared  with  the  hemisphere. 


REFRACTION     OF    LIGHT.  245 

But  there  are  curves  which  will  reflect  without  aberration. 
Let  a  concave  mirror  be  ground  to  the  form  of  a  paraboloid,  and 
rays  parallel  to  its  axis  will  be  converged  to  the  focus  without 
aberration.  For,  at  any  point  on  such  a  mirror,  a  line  parallel  to 
the  axis,  and  a  line  drawn  to  the  focus,  make  equal  angles  with 
the  tangent,  and  therefore,  equal  angles  with  the  perpendicular  to 
the  surface.  And  rays,  parallel  to  the  axis  of  a  convex  paraboloid, 
will  diverge  as  if  from  its  focus,  on  the  same  account.  Again,  if  a 
radiant  is  placed  at  the  focus  of  a  concave  parabolic  mirror,  the 
reflected  rays  will  be  parallel  to  the  axis,  and  will  illuminate  at  a 
great  distance  in  that  direction.  Such  a  mirror,  with  a  lamp  in 
its  focus,  is  placed  in  front  of  the  locomotive  engine  to  light  the 
track,  and  has  been  much  used  in  light-houses.  If  a  concave  mir- 
ror is  ellipsoidal,  light  emanating  from  one  focus  is  collected  with- 
out aberration  to  the  other,  because  lines  from  the  foci  to  any  point 
of  the  curve  make  equal  angles  with  the  tangent  at  that  point. 

Since  heat  is  reflected  according  to  the  same  law  as  light,  a 
concave  mirror  is  a  burning-glass.  When  it  faces  the  sun,  the 
light  and  heat  are  both  collected  in  a  small  image  of  the  sun  at 
the  principal  focus.  And,  if  no  heat  were  lost  by  the  reflection, 
the  intensity  at  the  focus  would  be  to  that  of  the  direct  rays,  as 
the  area  of  the  mirror  to  the  area  of  the  sun's  image.  Burning 
mirrors  have  sometimes  been  constructed  on  a  large  scale,  by  giv- 
ing a  concave  arrangement  to  a  great  number  of  plane  mirrors. 


CHAPTER   III. 

REFRACTION     OF     LIGHT. 

367.  Division  of  the  Incident  Beam.— When  light  falls 
on  an  opaque  body,  we  have  noticed  that  it  is  arrested,  and  a 
shadow  formed  beyond.     Of  the  light  thus  arrested,  a  portion  is 
reflected,  and  another  portion  lost,  which  is  said  to  be  absorbed  by 
the  body.     When  light  meets  a  transparent  body,  a  part  is  still 
reflected,  and  a  small  portion  absorbed,  but,  in  general,  the  greater 
part  is  transmitted.     The  ratio  of  intensities  in  the  reflected  and 
transmitted  beams  varies  with  the  angle  of  incidence,  but  little 
being  reflected  at  small  angles  of  incidence,  and  almost  the  whole 
at  angles  near  90°. 

368.  Refraction. — The  transmitted  beam  suffers  important 
changes,  one  of  which  is  a  change  in  direction.     This  change  is 


246 


OPTICS. 


FIG.  228. 


called  refraction9  and  takes  place  at  the  surface  of  a  new  medium. 
In  Fig.  228,  A  C9  incident  upon  R  S,  the  surface  of  a  different 

medium,  is  turned  at  C  into  another 
line,  as  C  E,  which  is  called  the  re- 
fracted ray.  The  angle  E  C  Q,  be- 
tween the  refracted  ray  and  the  perpen- 
dicular is  called  the  angle  of  refraction  ; 
the  angle  G  C  E,  between  the  direc- 
tions of  the  incident  and  the  refracted 
rays,  is  the  angle  of  deviation. 

It  is  a  general  fact,  to  which  there 
are  but  few  exceptions,  that  a  ray  of 
light  in  passing  out  of  a  rarer  into  a  denser  medium  is  refracted 
toward  the  perpendicular  to  the  surface ;  and  in  passing  out  of  a 
denser  into  a  rarer  medium,  it  is  refracted  from  the  perpendicular. 
But  the  chemical  constitution  of  bodies  sometimes  affects  their 
refracting  power.  Some  inflammable  bodies,  as  sulphur,  amber, 
and  certain  oils,  have  a  great  refracting  power  in  comparison  with 
other  bodies ;  and  in  a  given  instance,  a  ray  of  light  in  passing 
out  of  one  of  these  substances  into  another  of  greater  density  may 
be  turned  from  the  perpendicular  instead  of  toward  it.  In  the 
optical  use  of  the  words,  therefore,  denser  is  understood  to  mean, 
of  greater  refractive  power  ;  and  rarer  signifies,  of  less  refractive 
poiver.  In  Fig.  228,  the  medium  below  R  8  is  of  greater  refrac- 
tive power  than  that  above. 

Let  A  K  (Fig.  229)  represent  a  straight  rod,  the  lower  end  A 

being  beneath  the  surface 
of  water.  The  rays  which 
diverge  from  the  point  A 

B  S**  are  bent  from  the  perpen- 

dicular A  B.  The  ray  A  d, 
which,  if  prolonged,  would 
enter  the  eye,  is  by  refrac- 
tion bent  so  as  to  pass  be- 
low, while  the  ray  A  C  de- 
viates at  C  and  enters  the 
eye  as  though  coming  from  A',,  thus  giving  to  the  rod  the  bent 
appearance  noticed  in  an  oar  when  in  use. 

In  the  same  manner,  the  bottom  of  a  river  appears  elevated, 
and  diminishes  the  apparent  depth  of  the  stream.  Let  a  small 
object  be  placed  in  the  bottom  of  a  bowl,  and  let  the  eye  be  with- 
drawn till  the  object  is  hidden  from  view  by  the  edge  of  the  bowl. 
If  now  the  bowl  be  filled  up  with  water,  the  object  is  no  longer 
concealed,  for  the  light,  as  it  emerges  from  the  water,  is  bent  away 
from  the  perpendicular,  and  brought  low  enough  to  enter  the  eye. 


FIG.  229. 


LIMIT    OF    TRANSMISSION. 


247 


369.  Law  of  Refraction. — The  law  which  is  found  to  hold 

true  in  all  cases  of  common  refraction  is  this  : 

The  angles  of  incidence  and  refraction  are  on  opposite  sides  of 
the  perpendicular  to  the  surf  ace,  and,  for  any  given  media,  the  sines 
of  the  angles  have  a  constant  ratio  for  all  inclinations. 

For  example,  in  Fig.  230,  if  A  C  is  refracted  to  E,  then  a  C 
will  be  refracted  to  e,  so  that  A  D  :  E  F 
::  a  d  :  ef-,  and  if  the  rays  pass  out  in 
a  contrary  direction,  the  ratio  is  also 
constant,  being  the  reciprocal  of  the 
former,  viz.,  E  F  :  A  D  : :  e  f  :  a  d. 

This  constant  ratio  is  called  the 
Index  of  Refraction  and  is  found  by 
dividing  the  sine  of  the  angle  of  inci- 
dence by  the  sine  of  the  angle  of  refrac- 
tion. 

A  ray  perpendicular  to  the  surface, 
passing  in  either  direction,  is  not  refracted ;  for,  according  to  the 
law,  if  the  sine  of  one  angle  is  zero,  the  sine  of  the  other  must  be 
zero  also. 

The  following  table  gives  the  indices  of  refraction,  the  ray 
being  supposed  to  pass  from  a  vacuum  into  the  substance  ;  such 
indices  are  termed  absolute  indices : 


Diamond 2.450 

Carbon  disulphide 1.678 

Oil  of  cassia 1.630 

Flint  glass  (mean) 1.600 

Quartz 1.548 

Canada  Balsam...  1.540 


Crown  glass  (mean) 1 .530 

Alcohol 1.372 

Water 1.336 

Ice 1.309 

Air..  ..1.000294 


FIG.  231. 


370.  Limit  of  Transmission  from  a  Denser  to  a 
Rarer  Medium. — As  a  consequence  of  the  law  of  refraction, 
there  is  a  limit  beyond  which  a  ray  cannot  escape  from  a  denser 
medium.  Let  A  G  (Fig.  231)  be  the 
ray  incident  upon  the  rarer  medium 
RES.  It  will  be  refracted  from  the 
perpendicular  D  F  into  the  direction 
C  E,  so  that  A  D  is  to  E  F  in  a  constant 
ratio  (Art.  369).  If  the  angle  A  CD  be 
increased,  F  C  E  must  also  increase  till 
at  length  its  sine  equals  0  8. 

Suppose    the    denser  medium  to  be 
water  and  the  rarer  air,  then 

Sine  A  CD          1 


Sine  E  CF  ""  1.336 


nearly ; 


248 


OPTICS. 


hence,  sine  E  G  F=  1.336  x  sine  A  CD.  HE  CFbe  increased 
to  90°,  then  sine  90°  =  1  =  1.336  x  sine  A  CD,  from  which  we 
find  sine  A  C  D  =  .  7485,  the  angle  corresponding  to  which  is 
48°  -28'.  If  the  angle  of  incidence  be  greater  than  48°  28',  its  sine 
would  exceed  .7485,  and  therefore  the  sine  of  the  angle  in  air 
sluull  exceed  unity,  which  is  impossible.  Hence  it  follows,  that 
whenever  the  angle  of  incidence  is  greater  than  that  at  which  the 
sine  of  the  angle  of  refraction  becomes  equal  to  radius,  the  ray 
cannot  be  refracted  consistently  with  the  constant  ratio  of  the 
sines. 

This  is  proved  also  by  experiment ;  the  emerging  ray  increases 
its  angle  of  refraction  till  it  at  length  ceases  to  pass  out.  Beyond 
that  limit  all  the  incident  rays  are  reflected  from  the  inner  surface 
of  the  denser  medium ;  and  this  reflection  is  more  perfect  than 
any  external  reflection,  and  is  called  total  reflection. 

The  limiting  angle  for  diamond  is  24°  12',  and  its  great  bril- 
liancy, when  properly  cut,  is  due  to  numerous  internal  total  re- 
flections which  cause  the  light  to  emerge  in  different  directions. 

371.  Opacity  of  Mixed  Transparent  Media.— Light  in 
passing  from  a  medium  to  a  different  one,  is  partly  reflected  and 
partly  refracted ;  if  this  be  often  repeated  in  a  mixed  medium  no 
light  is  transmitted.     It  is  the  frequency  of  reflection  at  the 
limiting  surfaces  of  air  and  water  that  renders  foam  opaque.     So 
also   a   transparent  crystal,  when   crushed,  becomes  an  opaque 
powder.     If  the  powder  be  wetted  with  a  liquid  having  the  same 
refractive  index  as  the  crystal,  the  reflections  will  be  prevented  and 
transparency  will  result. 

372.  Transmission     through     Parallel    Plane    Sur- 
faces.— Let  S  (Fig.  232)  enter  the  medium  A,  and  represent  the 
emergent  ray  by  8'.     Suppose  the  ray  to  enter  from  a  vacuum, 
and  to  emerge  into  a  vacuum  again,  and  call  the  index  of  refrac- 
tion m.     Then 

sine  a  =  m  x  sine  a',  and 

sine  a'  =  —  sine  a"  ; 
m 

multiplying  these  together,  we  have   sine  a  =  sine  a",  whence 

a  =  a",  and  the  emergent  and  incident 
rays  are  parallel.  Suppose  the  ray  Sf 
to  enter  a  second  medium  B,  bounded 
by  parallel  faces,  it  will  emerge  parallel 
to  Sr,  and  therefore  parallel  to  S. 
Hence  if  a  ray  traverse  any  number 
of  media  with  parallel  faces,  these 
media  being  separated  by  vacua  -the 


FIG.  232. 


RELATIVE    INDICES    OF    REFRACTION.         249 


finally  emergent  ray  will  be  parallel  to  the  first  incident  ray  8. 
If  now  the  spaces  between  the  media  be  diminished,  the  result 
will  not  be  changed,  and  finally  when  the  diminution  reaches  its 
limit  the  faces  of  the  media  will  be  in  contact,  and  we  shall  still 
have  the  incident  and  emergent  rays  parallel. 

373.  Determination  of  Relative  Indices  of  Refrac- 
tion.— When  a  ray  passes  from  a  medium  A  into  another  B 
{Fig.  233),  the  absolute  indices  of  these  being  known  the  relative 
index  may  be  found.  Suppose  the  FlG  233. 

media  to  be  bounded  by  parallel  plane 
faces.  Let  m  be  the  absolute  index  of 
A,  and  n  that  of  B.  Denote  the 

,  , .      .    ,        sine  a'         .      0 

relative  index,  -; T  by  ^.     Suppose 

'  sine  a  ,    J 

the  ray  S  to  enter  A  from  a  vacuum, 
then 

sine  a  =  m  x  sine  a' 

sine  a'  =  i  x  sine  a" 


sine  a    =  -  sine 
n 


a  since    the  emergent  ray  S'  is  parallel 

to  the  .incident  ray  S  (Art.  372).     By  multiplying  these  equations 

n 

together,  we  find  i  =  — ;  hence,  to  find  the  relative  index  of  re- 
fraction when  a  ray  passes  from  medium  A  into  medium  B,  divide 
the  absolute  index  of  B  by  that  of  A. 

Suppose  a  ray  to  pass  from  air  into  carbon  disulphide,  then 

— fr-:—  =  1.6774,  knowing  which  the  deviation  of  the  ray  for  any 
l.UOOo 

given  angle  of  incidence  can  be  found. 

The  same  principle  may  be  applied  to  find  the  relative  index 
of  two  substances  whose  relative  indices  with  respect  to  a  third 
are  known. 

374.  Transmission  through  a  Medium  Bounded  by 
Inclined'  Planes. — A  medium  bounded  by  inclined  planes  is 


called  a  prism.  The  angle 
included  by  the  planes 
through  which  the  light 
passes  is  called  the  re- 
fracting angle  of  the 
prism,  and  the  planes  are 
deviating  planes. 

Let  A  B  (Fig.  234)  be 
the  incident  ray,  and  C  G 


FIG.  234. 


250  OPTICS. 

the  emergent  ray.  The  total  deviation  will  be  G  D  H  =  d. 
Adopting  the  notation  of  the  figure,  we  have  GDH  =  DBC  + 
D  C  B  or  d  —  (i  —  i')  -f  (e  —e')  =  i  +  e  —  (i1  -f  e').  Because 
of  the  perpendiculars  through  B  and  C  we  have  r  =  p,  but  p  =  i' 
-f  e'  =  r  ;  hence,  d  =  i  -{-  e  —  r  ;  that  is  to  say,  the  total  devia- 
tion is  equal  to  the  sum  of  the  angles  of  incidence  and  emergence 
diminished  by  the  refracting  angle  of  the  prism. 

375.  Prism  Used  for  Measuring  Refractive  Power.— 

For  any  given  prism  the  deviation  will  depend  upon  the  angles 
of  incidence  and  emergence. 

If  a  prism  rotate  about  an  axis  parallel  to  its  refracting  edge,  a 
position  of  minimum  deviation  will  be  found  such  that  any  rota- 
tion either  to  right  or  left  will  increase  the  deviation  of  the  ray  ; 
if  now  the  angles  of  incidence  and  emergence  be  measured,  they 
will  be  found  equal. 
From  the  equations 
r  =  i'  +  e' 

d  =  i  +  e  —  r,   by  making  i  =  e,  and  consequently 
t'  =  e',  we  obtain 

t  =  i(r  +  d),  andi'  =  ^r; 
from  which  we  find  the  relative  index  of  refraction 
sine  i        sine  £  (r  -\-  d) 

171  mi  — : Ti    mi    : 1 . 

sine  i  sine  %  r 

Thus  having  measured  the  refracting  angle  of  the  prism  and 
the  minimum  deviation  of  the  ray  we  can  at  once  determine 
the  index  of  refraction  of  the  substance  of  which  the  prism  is 
formed. 

If  the  angle  r  be  very  small,  d  will  also  be  small,  and  the  ratio 
of  the  angles  may  be  used  instead  of  the  ratio  of  their  sines,  and 

the  formula  then  becomes  m  = =  1-1 — . 

r  r 

This  is  one  of  the  best  methods  by  which  to  determine  the 
index  of  refraction  of  a  solid,  transparent  substance. 

The  final  deviation  of  the  ray  being  unaffected  by  its  passage 
through  glass  plates  with  parallel  faces,  hollow  prisms  formed  of 
such  plates  may  be  filled  with  a  liquid  whose  index  of  refraction 
is  to  be  determined.  A  tube  whose  end  sections  are  glass  planes 
equally  inclined  to  the  axis  of  the  tube,  may  be  used  to  determine 
the  relative  indices  of  gases  and  air,  and  by  exhausting  the  tube 
to  form  a  vacuum,  the  absolute  indices  may  be  found. 

376.  Light  through  One  Surface.— 

1.  Plane  Surface.  When  parallel  rays  pass  into  another  me- 
dium through  a  plane  surface,  they  remain  parallel.  For  the  per- 


LIGHT    THROUGH    ONE    SURFACE. 


251 


pendiculars  being  parallel,  the  angles  of  incidence  are  equal,  and 
therefore  the  angles  of  refraction  are  equal  also,  and  the  refracted 
rays  parallel.  But  a  pencil  of  diverging  rays  is  made  to  diverge 
less,  when  it  enters  a  denser  medium.  For  the  outer  rays  make 
the  largest  angles  of  incidence,  and  are  therefore  most  refracted 
toward  the  perpendiculars,  and  thus  toward  parallelism  with  each 
other.  And  when  diverging  rays  enter  a  rarer  medium,  they  di- 
verge more  ;  because  the  outside  rays  make  the  largest  angles  of 
incidence,  and  therefore  the  largest 
angles  of  refraction,  by  which  means 
they  spread  more  from  each  other. 

The  last  case  is  illustrated  when  we 
look  perpendicularly  into  water,  and 
see  its  depth  apparently  diminished  by 
about  one-fourth  of  the  whole.  Let 
A  B  (Fig.  235)  be  the  surface,  and  C  a 
point  at  the  bottom,  from  which  pen- 
cils come  to  the  eyes  at  E,  E'.  Let  GF 
be  perpendicular  to  the  surface  A  B, 
and  C  B  E  the  axis  of  an  oblique  pen- 
cil to  the  eye  at  E.  As  the  distance 
between  the  pupils  of  the  eyes  is  less 
than  2-J-  inches,  the  obliquity  of  the 
pencil  C  B  E  will  be  very  slight.  ° 

The  angle  C  =  G  B  H  =  angle  of  incidence  ;  and  A  D  B  = 
G  B  E  =  angle  of  refraction.  Now,  in  the  triangle  B  D  C,  B  C: 
B  D  (: :  A  C  :  A  D  nearly)  : :  sin  D  :  sin  C  : '  sine  of  refraction  : 
sine  of  incidence::  1.34  :  1.  Hence  the  apparent  depth  is  one 
fourth  less  than  the  real  depth.  The  apparent  depth  of  water  may 
be  diminished  much  more  than  this  by  looking  into  it  obliquely. 

2.  Convex  surface  of  the  denser.  A  convex  surface  tends  to 
converge  rays.  Let  0'  (Fig.  236)  be  the  centre  of  convexity,  and 

FIG.  236. 


C'  D,  G'  (7,  two  radii  produced.  As  rays  are  bent  toward  the  per- 
pendiculars in  entering  a  denser  medium,  and  as  the  perpendicu- 
lars themselves  converge  to  (7',  the  general  effect  of  such  a  surface 
is  to  produce  convergency.  The  pencil,  A  H,  A  JV,  is  merely 
made  less  divergent,  H  D'  N  A1 ;  B  H,  B  N  become  parallel, 


252 


OPTICS. 


II D',  N  B' ;  D  H,  D  N,  convergent  to  D' ;  the  parallel  rays,  D  H, 
EN,  convergent  to  E' ;  the  convergent  pencil,  D  H,  F N,  more 
convergent  to  F' ;  but  D  H,  C  N,  which  converge  equally  with 
the  radii,  are  not  changed ;  and  D  H,  G  N,  which  converge  more 
than  the  radii,  converge  less  than  before,  to  G'.  The  two  last 
cases,  which  are  exceptions  to  the  general  effect,  rarely  occur  in 
the  practical  use  of  lenses. 

If  we  trace  in  the  opposite  direction  the  rays,  A',  B\  D',  &c., 
comparing  each  with  D'  D,  we  find,  in  this  case  also,  that  the 
convex  surface  tends  to  converge  the  rays,  by  bending  them  from 
their  respective  perpendiculars. 

3.  Concave  surface  of  the  denser.  A  concave  surface  tends  to 
diverge  rays.  Let  C  C',  CD  (Fig.  237),  be  the  radii  of  concavity 
produced.  As  the  radii  diverge  in  the  direction  in  which  the  light 

FIG.  237. 


moves,  the  rays,  being  bent  toward  them,  will  generally  be  made 
to  diverge  also.  Hence,  parallel  rays,  B  H,  E  N,  are  diverged, 
H  D,  N  E' ;  and  diverging  rays,  B  H,  B  N,  are  diverged  more, 
H  D,  N  B'.  If,  however,  rays  diverge  as  much  as  the  radii,  or 
more,  they  proceed  in  the  same  direction,  or  diverge  less,  a  case 
which  rarely  occurs. 

If  the  rays  are  traced  in  the  opposite  direction,  the  tendency 
in  general  to  produce  divergency  appears  from  the  fact  that  the 
perpendiculars  are  now  converging  lines,  and  the  rays  are  refracted 
from  them. 

^l 

377.  Lenses.— A  lens  is  a  transparent  medium  bounded  by 
curved  surfaces  whose  centres  of  curvature  lie  upon  a  normal 
common  to  the  two  surfaces.  If  the  radius  of  curvature  is  made 

FIG.  288. 


infinite,  the  corresponding  surface  becomes  a  plane.     The  usual 
varieties  are  shown  in  Fig.  238. 


CONVEX     AND    CONCAVE     LEXS.  353 

A  double  convex  lens  (A)  consists  of  two  spherical  segments, 
either  equally  or  unequally  convex,  having  a  common  base. 

A  plano-convex  lens  (B)  is  a  lens  having  one  of  its  sides  convex 
and  the  other  plane,  being  simply  a  segment  of  a  sphere. 

A  double  concave  lens  (C)  is  a  solid  bounded  by  two  concave 
spherical  surfaces,  which  may  be  either  equally  or  unequally  con- 
cave. 

A  plano-concave  lens  (D)  is  a  lens  one  of  whose  surfaces  is 
plane  and  the  other  concave. 

A  meniscus  (E)  is  a  lens  one  of  whose  surfaces  is  convex  and 
the  other  concave,  but  the  concavity  being  less  than  the  convexity, 
it  takes  the  form  of  a  crescent,  and  has  the  effect  of  a  convex  lens 
whose  convexity  is  equal  to  the  diiference  between  the  sphericities 
of  the  two  sides. 

A  concavo-convex  lens  (F)  is  a  lens  one  of  whose  surfaces  is  con- 
vex and  the  other  concave,  the  concavity  exceeding  the  convexity, 
and  the  lens  being  therefore  equivalent  to  a  concave  lens  whose 
concavity  is  equal  to  the  difference  between  the  sphericities  of  the 
two  sides. 

A  line  (M  N)  passing  through  a  lens,  perpendicular  to  its  op- 
posite surfaces,  is  called  the  axis.  The  axis  usually,  though  not 
necessarily,  passes  through  the  centre  of  the  figure. 

378.  General  Effect  of  the  Convex  Lens.— Whether 
double-convex  or  plano-convex,  its  general  effect  is  to  converge  light. 
It  has  been  shown  (Art.  376)  that  the  convex  surface  of  a  denser 
medium  tends  to  converge  rays,  whichever  way  they  pass  through 
it.  Therefore,  if  E  (Fig.  239)  is  a  radiant,  while  E  C'  C  follows 

FIG.  239. 


the  axis  without  change  of  direction,  the  oblique  ray  E  D  is  first 
refracted  toward  D  C,  and  then  from  C'  D'  produced,  and  both 
actions  conspire  to  converge  it  to  the  axis.  The  rays  are  repre- 
sented as  meeting  in  the  focus  F.  Whether  the  rays  are  actually 
converged,  depends  on  their  previous  relation  to  each  other.  If 
the  lens  is  plano-convex,  the  plane  surface  has  usually  but  little 
effect  in  converging  the  light ;  but  by  Art.  376  it  may  be  shown 
that  its  action  will  usually  conspire  with  that  of  the  convex  sur- 
face. 


254  OPTICS. 

379.  General  Effect  of  the  Concave  Lens.— This  Jens, 
whether  double-concave  or  plano-concave,  tends  to  produce  diver- 
gency. This  is  evident  from  what  has  been  shown  in  Art.  376. 
The  ray  E  D  (Fig.  240),  in  entering  the  denser  medium,  is  first 

FIG.  240. 


refracted  toward  0'  D  produced,  and  on  leaving  the  medium  at  D't 
is  refracted  from  D'  (7;  and  is  thus  twice  refracted  from  the  ray 
E  C,  which  being  in  the  axis,  is  not  refracted  at  all.  If  the  lens 
is  plano-concave,  the  eff ect  of  the  plane  surface  may,  or  may  not, 
conspire  with  that  of  the  concave  surface. 

380.  The  Optic  Centre  of  a  Lens. — The  incident  and 
emergent  portions  of  a  ray  which  enters  and  leaves  a  lens  at  the 
points  of  contact  of  parallel  tangent  planes  will  be  parallel 
according  to  Art.  372. 

The  point  where  the  part  of  such  ray  included  between  the 
bounding  surfaces  cuts  the  axis  of  the  lens,  or  would  cut  it  if  pro- 
duced, is  called  the  optic  centre. 

In  Fig.  241  let  a  and  b  be  points  of  contact  of  parallel  tangent 

planes,  then  the  radii  C  a 
and  C'  b  being  perpendicu- 
lar to  these  parallel  planes 
are  themselves  parallel, 
hence  the  angles  o  and  o 
are  equal ;  the  angles  at 
P  are  also  equal,  and  hence 
the  triangles  C  a  P  and 
O'  b  P  are  similar,  and 
C'  P  :CP::  C'  b:  Ca. 

i 

Represent  the  thickness 

of  the  lens  x  y,  measured  on  the  axis,  by  #,  and  the  distance  from 
P,  the  optic  centre,  to  the  surface  x  by  e ;  also  make  the  radius 
C  a  =  r  and  O'  b  =  r.  Substituting  these  values  above  we 
have 

r'  _  e  :  r  —  (t  —  e}  : :  r'  :  r, 
from  which  we  obtain 

r't  r' 


e  = 


r    .     r 
1  ~~  ~t 


CONJUGATE    FOCI.  255 

But  this  value  of  e  is  constant  since  r,  r'  and  t  are  constant ; 
therefore  all  rays  which  suffer  no  deviation  in  passing  through 
the  lens  must  pass  through  a  common  point  P,  called  the  optic 
centre.  The  optic  centre  is  within  the  lens  in  the  cases  of  double 
concave  and  double  convex  lenses,  but  without  in  the  meniscus 
and  concavo-convex.  If  r  =  r'  the  optic  centre  is  midway  between 
the  faces. 

381.  Conjugate  Foci.— If  the  rays  from  R  (Fig.  24=2)  are 
collected  at  F,  then  rays  emanating  from  F  will  be  returned 
to  R\  and  the  two  points  are  called  conjugate  foci.  Their  relative 
distances  from  the  lens  may  be  determined  when  the  radii  of  the 

FIG.  242. 


w  —  1:1 

n  —  1:1 


surfaces  and  the  index  of  refraction  are  known.  Let  n  be  the 
index  of  refraction,  and  assume,  what  is  practically  true,  that  the 
angles  of  incidence  and  refraction  are  so  small  that  their  ratio  is 
the  same  as  the  ratio  of  their  sines.  Then 

RQ  P(=KG  I)  :/  O  H: 

A      KGH'.IG  H: 

in  like  manner  K  H  G  :  I  H  G  : 

.:KGH+  KH  G  :  I  G  H  +  I  H  G  ::  n  -  1:1. 
But      KGH+KHG  =  BKF=R  +  F; 
and      IGH+  IHQ=GI-C=C  +  C'; 
naming  the  acute  angles  at  R,   C,    O'  F,  by  those  letters  re- 
spectively, 

.:  R  +  F:C  +  C'::n-l:l. 

Now,  the  lens  being  thin,  and  the  angles  R,  C,  C',  and  F  very 
small,  the  same  perpendicular  to  the  axis,  at  L,  the  centre  of  the 
lens,  may  be  considered  as  subtending  all  those  angles.  Hence, 
each  angle  is  as  the  reciprocal  of  its  distance  from  L.  Let  R  L  — 
P',  F  'L  =  q;  C  L  =  r  ;  and  O'  L  =  r'.  Then  the  equation  above 
becomes, 


which  expresses  in  general  the  relation  of  the  conjugate  foci. 

382.  To  Find  the  Principal  Focus.—  The  radiant  from 
which  parallel  rays  come  is  at  an  infinite  distance.     Therefore, 


256  OPTICS. 

making  p  =  x  ,  and  the  distance  of  the  principal  focus  =  F,  we 

have  -  =  0,  and 

111 

_:_+_::,  _1:1. 

If  the  curvatures  are  equal,  for  crown-glass,  for  which  n  =  [f 

F  reduces  to  r  ;  that  is,  the  principal  focus  of  a  double  convex 
lens  of  crown-glass,  having  equal  curvatures,  is  at  the  centre  of 
convexity. 

The  foregoing  formulae  are  readily  adapted  to  the  other  forms 
of  lens.  When  a  surface  is  plane,  its  radius  is  infinite,  and 

-  ,  or  —  =  0.     When  concave,  its  centre  is  thrown  upon  the  same 

side  as  the  surface,  and  its  radius  is  to  be  called  negative.  And  if 
the  focal  distance,  as  given  by  the  formula,  becomes  negative, 
it  is  understood  to  be  on  the  same  side  as  the  radiant;  that  is, 
the  focus  is  a  virtual  radiant. 

383.  Powers  of  Lenses  Practically  Determined.—  The 
reciprocal  of  the  principal  focal  length  of  a  lens  -^,  is  called  the 
power  of  a  lens.  From  Art.  381  we  find 

H-<—  >€*&••,    I 

and  from  Art.  382       \ 


whence  we  have 


As  the  index  of  refraction  and  the  radii  of  curvature  are  not 
generally  known  in  respect  to  any  particular  lens  which  we  may 
happen  to  be  using,  some  practical  method  by  which  to  determine 
F  will  enable  us  to  calculate  readily  either  p  or  q,  the  other  being 
given. 

(1.)  To  find  F  for  a  convex  lens.  —  Form  an  image  of  the  sun 
upon  a  plate  of  ground  glass,  and  measure  the  distance  of  the 
image  from  the  lens.  Or,  place  a  light  on  one  side  of  the  lens 
and  find  its  sharp  image  upon  a  screen  on  the  other  side.  These 

distances  measured,  give  p  and  q,  whence  F  =  . 

(2.)  These  two  methods  assume  the  thickness  of  the  lens  to  be 
small  compared  with  the  focal  length.  The  focal  length  of  a 


EQUIVALENT    COMBINATIONS.  35? 

thick  lens,  or  system  of  lenses,  may  be  found  thus :  On  one  side, 
at  a  distance  a  little  greater  than  F}  place  a  scale  strongly  illumi- 
nated by  transmitted  light,  and  receive  the  sharp  and  greatly 
magnified  image  of  one  of  its  divisions  upon  a  screen  upon  the 
other  side  of  the  lens  or  lenses.  Then  let  I  =  length  of  one 
division,  L  =  length  of  its  image,  p  =  distance  of  the  screen 
from  the  lens  (very  great  compared  with  its  thickness),  and  we 

find,  from  similar  right-angled  triangles,  L  :  I ::  p  :  £-=-  =  q,  and 
these  yalues  of  p  and  q  give 

PP~ 

~  p  +  q~p~+  pi 
L 

The  focal  length  is  strictly  the  distance  from  F  to  the  intersection 
of  the  axis  by  the  principal  plane  of  the  lens  or  combination  of 
lenses. 

The  principal  plane  passes  through  the  point  of  intersection 
of  an  incident  ray,  parallel  to  the  axis  and  its  emergent  ray,  both 
produced  if  necessary,  and  is  at  right  angles  to  the  axis. 

(3.)  To  find  F  for  a  concave  lens.  Use  in  contact  with  the 
concave  lens  a  stronger  convex,  of  known  value  for  F,  and  pro- 
ceed according  to  the  preceding  methods.  Then  if  f  =  focal 
length  of  combination  and  /'  =  focal  length  of  convex  alone,  and 
F  =  that  of  concave  lens  sought,  we  shall  find 

•p=  *  —  f>  >  as  will  be  proved  hereafter  ;  or  when  the  lens  is  deep 

J       J 

and  not  very  small,  take  for  the  focal  length  that  distance  from 
a  screen  at  which  the  circle  of  light  from  the  sun  is  twice  the 
diameter  of  the  lens. 

384.  Equivalent  Combinations. — 

To  find  the  focal  length  of  a  lens  which  shall  be  equal  to  a  com- 
bination of  two  lenses. 

Suppose  the  lenses  (Fig.  243)  to  be  of  such  thickness  as  may 
be  neglected.  Let  a 

ray    parallel    to    the  FIG.  243. 

common  axis  be  inci- 
dent at  R.  If  R  V 
be  drawn  parallel  to 
8  T,  the  emergent 
ray,  A  V  will  repre- 
sent the  focal  length,  F9  of  a  lens  which  would  produce  the  same 
deviation  as  this  combination.  Let  A  X  =  /  =  focal  length  of 
A,  then  B  X  =  f  —  a,  a  being  the  distance  between  B  and  A. 
17 


258 


OPTICS. 


Now  if  we  regard  T  as  a  radiant,  and  T  S  R  as  the  path  of  the 
ray,  then  X  is  the  virtual  conjugate  focus  of  the  lens  B  corre- 
sponding to  T,  and  calling  /'  the  focal  length  of  B,  we  have 

jnirt.    OOOf   ~p~t    —  -jz    -fp  ~~~  ~j^    «.. 

Substituting  the  value  of  B  X  above,  we  have 

J_       1        JL_    _/'+/-* 

B  T~  /'  "V  —  a  ~  /'(/  —  a) ' 

By  similar  triangles  A  V  R,  B  T  S  and  X  A  R,  X  B  S 

BT  B S       B  X      , 

A   V(--pi\  =  A~7?=  ~A~X>  Whence 

^1     V    I    —  J?    1  ^a.    It          JL  A 


When  the  lenses  are  in  contact  the  distance  a  =  o,  and  we  have 

-=  =--+_;  that  is  to  say, 
*        J      / 

TJiepoiuer  of  a  combination  of  two  lenses  in  contact  is  equal  to 
the  sum  of  their  respective  powers. 

Due  attention  must  be  paid  to  the  signs  of  the  powers,  those 
of  concave  lenses  being  negative. 

The  above  rule  is  general,  and  is  not  confined  to  two  lenses 
only. 

385.  Images  by  the  Convex  Lens. — The  convex  lens 
forms  a  variety  of  images,  whose  character  and  position  depend  on 
the  place  of  the  object.  If  it  is  at  the  principal  focus,  the  rays  of 
every  pencil  pass  out  parallel,  and  seem  to  come  from  an  infinite 
distance.  If  the  object  is  nearer  than  the  principal  focus,  the 
emergent  rays  of  each  pencil  diverge  less  than  the  incident  rays, 
and  therefore  seem  to  radiate  from  points  further  back  ;  the  image 
is  therefore  apparent. 

Let  M  N  be  an  object  (Fig.  244)  nearer  than  the  principal 

FIG.  244. 


IMAGES    BY    THE    CONVEX    LENS.  359 

focus.  Kays  from  the  point  M  will  diverge ;  and  falling  upon 
the  whole  surface  of  the  lens,  will  be  refracted ;  some  passing 
above  the  eye  and  some  below ;  only  a  small  part  of  them  wiJl 
enter  the  eye,  as  though  they  came  from  m,  situated  on  the 
secondary  axis  G  M  produced.  In  like  manner  each  point  of 
M  N  has  its  corresponding  point  in  the  image  m  n. 

The  image  is  erect,  because  the  axes  of  the  pencils  do  not  cross 
between  the  object  and  image  ;  and  it  is  enlarged,  because  it  sub- 
tends the  angle  M  C  JVat  a  greater  distance  than  the  object. 

But  if  the  object  is  further  from  the  lens  than  the  principal 
focus,  the  rays  of  each  pencil  converge  to  a  point  in  the  axis  of 
that  pencil  produced  through  the  lens;  and  thus  light  is  collected 
in  focal  points,  which  consequently  become  actual  radiants. 

Let  M  N  (Fig.  245)  be  the  object.  A  cone  of  rays  from  the 
point  M  covers  the  lens,  and  converges  to  the  conjugate  focus 

FIG.  245. 


m,  on  the  axis  M  C  produced,  whence  the  rays  again  diverge  as 
from  a  real  radiant.  Some  of  these,  as  m  c,  pass  above  the  eye, 
while  others,  ml,  m  V,  m  b",  pass  below,  only  a  small  part  of 
them  entering  the  eye  and  rendering  visible  the  image  m  n. 

Instead  of  viewing  the  image  as  above,  since  it  is  a  collection 
of  real  foci  a  white  screen  may  be  placed  at  m  n,  which  will  re- 
flect the  light  of  each  focal  point  in  all  directions,  and  thus  render 
the  image  visible  to  a  large  audience.  Though  the  rays  of.  every 
radiant  converge  from  the  lens  to  the  conjugate  focus  of  that 
radiant,  yet  the  axes  of  the  pencils  diverge  from  each  other,  having 
all  crossed  at  the  optic  centre.  The  image  is  therefore  inverted, 
as  are  all  real  images,  in  whatever  way  produced. 

The  formula  for  conjugate  foci  shows  that  if  p  is  increased,  q 
is  diminished  ;  therefore  the  further  M  N  is  removed  from  the 
lens,  the  nearer  m  n  approaches  to  it ;  but  the  nearest  position  is 
the  principal  focus,  which  it  reaches  when  the  object  is  at  an  infi- 
nite distance.  As  the  object  and  image  subtend  equal  angles  at 
the  optic  centre,  and  are  parallel,  or  nearly  parallel  with  each 
other,  their  diameters  are  proportional  to  their  distances  from  the 


260 


OPTICS. 


lens.  But  the  area  of  the  lens  has  no  effect  on  the  size  of  the 
image,  since  change  of  area  does  not  alter  the  relation  of  the  axes, 
but  only  the  size  of  the  luminous  cones,  and  thus  the  quantity  of 
light  in  each  pencil. 

386.  Images  by  the  Concave  Lens. — As  the  rays  of  each 
pencil  are  diverged  more  after  passing  through  the  lens  than  before, 
the  image  is  apparent,  and  is  situated  between  the  lens  and  the 
object.  Let  M  N  (Fig.  246)  be  the  object ;.  the  cone  of  rays  from 
JVwill,  after  refraction,  diverge  more,  as  from  n,  in  the  same  axis 

FIG.  246. 


c  N;  and  all  other  pencils  will  be  affected  in  a  similar  manner, 
and  form  an  apparent  image  m  n.  It  will  be  erect,  since  the  axes 
do  not  cross  between,  and  diminished,  being  nearer  the  angle  c, 
which  is  subtended  by  both  object  and  image. 

It  is  noticeable  that  the  concave  mirror  and  the  convex  lens  are 
analogous  in  their  effects,  forming  images  on  both  sides,  both  real 
and  apparent,  both  erect  and  inverted,  both  larger  and  smaller 
than  the  object ;  while  the  convex  mirror  and  the  concave  lens  also 
resemble  each  other,  producing  images  always  on  one  side,  always 
apparent,  always  erect,  always  smaller  than  the  object. 

387.  Caustics  by  Refraction.— If  the  convex  surface  of  a 
lens  is  a  considerable  part  of  a  hemisphere,  the  rays  more  distant 
from  the  axis  will  be  so  much  more  refracted  than  others,  as  to 

cross  them  and  meet  the  axis  at  nearer 
points,  thus  forming  caustics  by  refrac- 
tion. Fig.  247  shows  this  effect  in  the 
case  of  parallel  rays ;  those  near  the 
axis  intersecting  it  at  the  principal 
focus  F,  and  the  intersections  of  re- 
moter rays  being  nearer  and  nearer  to 
the  lens,  so  that  the  whole  converging 
pencil  assumes  a  form  resembling  a  cone  with  concave  sides. 

The  grating  (Fig.  248),  viewed  through  such  a  lens,  would 


FIG.  247. 


SPHERICAL    ABERRATION. 


appear  distorted,  as  in  Fig.  249,  and  if  viewed  through  a  con- 
cave lens  the  opposite  effect  would  result,  as  in  Fig.  250. 

FIG.  248.  FIG.  249.  FIG.  250. 


388.  Spherical  Aberration  of  a  Lens. — The  production 
of  caustics  is  an  extreme  case  of  what  is  called  spherical  aberra- 
tion.    Unless  the  lens  is  of  small  angular  breadth,  not  more  than 
10°,  a  pencil  whose  rays  originated  in  one  point  of  an  object 
is  not  converged  accurately  to  one  point  of   the  image,  but 
the  outer  rays  are   refracted  too  much,   and  make  their  focus 
nearer  the  lens  than  that  of  the  central  rays,  as  represented  in 
Fig.  251.     If  F  is  the  focus 

of  the  central  rays,  and  F'  FIG.  251. 

of  the  extreme  ones,  other 

rays  of  the  same  beam  ure 

collected    in    intermediate 

points,  and  F  F'  is  called 

the  longitudinal   spherical 

aberration ;  and   G  H,  the 

breadth  covered  by  the  pencil  at  the  focus  of  central  rays,  is  called 

the  lateral  spherical  aberration. 

Such  a  lens  cannot  form  a  distinct  image  of  any  object ;  be- 
cause perfect  distinctness  requires  that  all  rays  from  any  one  point 
of  the  object  should  be  collected  to  one  point  in  the  image.  If, 
for  example,  the  beam  whose  outside  rays  are  R  A,  R  B,  cornea 
from  a  point  of  the  moon's  disc,  that  point  will  not  be  perfectly 
represented  by  F,  because  a  part  of  its  light  covers  the  circle, 
whose  diameter  is  G  H,  thus  overlapping  the  space  representing 
adjacent  points  of  the  moon.  And  if  that  point  had  been  on  the 
edge  of  the  moon's  disc,  F  could  not  be  a  point  of  a  well-defined 
edge  of  the  image,  since  a  part  of  the  light  would  be  spread  over 
the  distance  F  G  outside  of  it,  and  destroy  the  distinctness  of  its 
outline. 

389.  Remedy  for  Spherical  Aberration.— As  spherical 
lenses  refract  too  much  those  rays  which  pass  through  the  outer 
parts,  it  is  obvious  that,  to  destroy  aberration,  a  lens  is  required 
whose  curvature  diminishes  toward  the  edges.    Accordingly,  forms 
for  ellipsoidal  lenses  have  been  calculated,  which  in  theory  will 
completely  remove  this  species  of  aberration.      But   no   curved 


262  OPTICS. 

solids  can  be  so  accurately  ground  as  those  whose  curvature  is  uni- 
form in  all  planes,  that  is,  the  spherical.  Hence,  in  practice  it  is 
found  better  to  reduce  the  aberration  as  much  as  possible  by  spher- 
ical lenses,  than  to  attempt  an  entire  removal  of  it  by  other  forms 
which  cannot  be  well  made. 

Lenses,  or  combinations  which  are  free  from  spherical  aberra- 
tion, are  said  to  be  Aplanatic.  By  lessening  the  aperture  of  a  lens 
by  a  suitable  diaphragm,  the  aberration  maybe  much  diminished. 

In  a  plano-convex  lens,  whose  plane  surface  is  toward  the  ob- 
ject, the  spherical  aberration  is  4.5  ;  that  is  (Fig.  251),  F  F1  =  4.5 
times  the  thickness  of  the  lens.  But  the  same  lens,  with  its  con- 
vex side  toward  the  object,  is  far  better,  its  aberration  being  only 
1.17.  In  a  double  convex  lens  of  equal  curvatures,  the  aberration 
is  1.67  ;  if  the  radii  of  curvature  are  as  1  :  6,  and  the  most  convex 
side  is  toward  the  object,  the  aberration  is  only  1.07.  By  placing 
two  plano-convex  lenses  near  each  other,  the  aberration  may  be 
still  more  reduced. 

390.  Atmospheric  Refraction. — The  atmosphere  may  be 
regarded  as  a  transparent  spherical  shell,  whose  density  increases 
from  its  upper  surface  to  the  earth.     The  radii  of  the  earth  pro- 
duced are  the  perpendiculars  of  all  the  laminae  of  the  air ;  and 
rays  of  light  coming  from  the  vacuum  beyond,  if  oblique,  are  bent 
gradually  toward  these  perpendiculars ;   and  therefore  heavenly 
bodies  appear  more  elevated  than  they  really  are.    The  greatest 
elevation  by  refraction  takes  place  at  the  horizon,  where  it  is  about 
half  a  degree. 

391.  Mirage. — This  phenomenon,  called  also  looming,  con- 
sists of  the  formation  of  one  or  more  images  of  a  distant  object, 
caused  by  horizontal  strata  of  air  of  very  different  densities.     Ships 
at  sea  are  sometimes  seen  when  beyond  the  horizon,  and  their 
images  occasionally  assume  distorted  forms,  contracted  or  elon- 
gated in  a  vertical  direction.     These  effects  are  generally  ascribed 
to  extraordinary  refraction  in  horizontal  strata,  whose  difference 
of  density  is  unusually  great.     But  many  cases  of  mirage  seem  to 
be  instances  of  total  reflection  from  a  highly  rarefied  stratum  rest- 
ing on  the  earth.     These  occur  frequently  on  extended  sandy 
plains,  as  those  of  Egypt.    When  the  surface  becomes  heated,  dis- 
tant villages,  on  more  elevated  ground,  are  seen  accompanied  by 
their  images  inverted  below  them,  as  in  water.     As  the  traveler 
advances,  what  appeared  to  be  an  expanse  of  water  retires  before 
him.     By  placing  alcohol  upon  water  in  a  glass  vessel,  and  allow- 
ing them  time  to  mingle  a  little  at  their  common  surface,  the 
phenomena  of  mirage  may  be  artificially  represented. 


THE    PRISMATIC    SPECTRUM. 


CHAPTER    IV. 


DECOMPOSITION  AND  DISPERSION  OP  LIGHT. 

392.  The  Prismatic  Spectrum. — Another  change  which 
light  suffers  in  passing  into  a  new  medium,  is  called  decomposi- 
tion, or  the  separation  of  light  into  colors.  For  this  purpose,  the 
glass  prism  is  generally  employed.  It  is  so  mounted  on  a  jointed 
stand,  that  it  can  he  placed  in  any  desired  position  across  the 
beam  from  the  heliostat.  The  beam,  as  already  noticed,  is  bent 
away  from  the  refracting  angle,  both  in  entering  and  leaving  the 
prism,  and  deviates  several  degrees  from  its  former  direction.  If 
the  light  is  admitted  through  a  narrow  aperture,  F  (Fig.  252),  and 

FIG.  252. 


the  axis  of  the  prism  is  placed  parallel  to  the  length  of  the  aper- 
ture, the  light  no  longer  falls,  as  before,  in  a  narrow  line,  L,  but 
is  extended  into  a  band  of  colors,  R  V,  whose  length  is  in  a  plane 
at  right  angles  to  the  axis  of  the  prism.  This  is  called  the  pris- 
matic spectrum.  Its  colors  are  usually  regarded  as  seven  in  num- 
ber— red,  orange,  yellow,  green,  blue,  indigo,  violet.  The  red 
is  invariably  nearest  to  the  original  direction  of  the  beam,  and 
the  violet  the  most  remote ;  and  it  is  because  the  elements  of 
white  light  are  unequally  refrangible,  that  they  become  separated, 
by  transmission  through  a  refracting  body.  The  spectrum  is 
properly  regarded  as  consisting  of  innumerable  shades  of  color. 
Instead  of  Newton's  division  into  seven  colors,  many  choose  to 
consider  all  the  varieties  of  tint  as  caused  by  the  combination  of 
three  primitive  colors,  red,  yellow,  and  blue,  varying  in  their  pro- 


264 


OPTICS. 


portions  throughout  the  entire  spectrum.  The  number  seven,  as 
perhaps  any  other  particular  number,  must  be  regarded  as  arbi- 
trary. 

The  spectrum  contains  rays  of  other  wave  lengths  than  those 
which  affect  the  eye.  The  rays  of  longest  wave  length  are  crowded 
together  at  and  beyond  the  red,  and  here  the  greatest  heat  is  found 
upon  testing  with  a  thermometer. 

The  chemical  or  actinic  rays  of  shortest  wave  lengths,  are 
found  at  and  beyond  the  violet.  These  invisible  rays  differ  from 
those  which  are  visible  only  in  wave  length. 

Light  from  other  sources  is  also  susceptible  of  decomposition 
by  the  prism ;  but  the  spectrum,  though  resembling  that  of  the 
sun,  usually  differs  in  the  proportion  of  the  colors. 

393.  The  Individual  Colors  of  the  Spectrum  cannot  be 
Decomposed  by  Refraction. — If  the  spectrum  formed  by  the 
prism  A  be  allowed  to  fall  on  the  screen  E  D  (Fig.  253),  and  one 
color  of  it,  green  for  example,  be  let  through  the  screen,  and 

FIG.  253. 


received  on  a  second  prism,  B,  it  is  still  refracted  as  before,  but 
all  its  rays  remain  together  and  of  the  same  color.  The  same  is 
true  of  every  color  of  the  spectrum.  Therefore,  so  far  as  re- 
frangibility  is  concerned,  all  the  colors  of  the  spectrum  are  alike 
simple. 

394.  Colors  of  the  Spectrum  Recombined.— It  may  be 

shown,  in  several  ways,  that  if  all  the  colors  of  the  spectrum  be 
combined,  they  will  reproduce  white  light.  One  method  is  by 
transmitting  the  beam  successively  through  two  prisms  whose 
refracting  angles  are  on  opposite  sides.  By  the  first  prism,  the 
colors  are  separated  at  a  certain  angle  of  deviation,  and  then  fall 
on  the  second,  which  tends  to  produce  the  same  deviation  in  the 
opposite  direction,  by  which  means  all  the  colors  are  brought  upon 
the  same  ground,  and  the  illuminated  spot  is  white  as  if  no  prism 
had  been  interposed.  Or  the  colors  may  be  received  on  a  series  of 
small  plane  mirrors,  which  admit  of  such  adjustment  as  to  reflect 
all  the  beams  upon  one  spot.  Or  finally,  the  several  colors  can, 
by  different  methods,  be  passed  so  rapidly  before  the  eye  that  their 


COMPLEMENTARY    COLOR'S.  265 

visual  impressions  shall  be  united  in  one ;  in  which  case  the  illu- 
minated surface  appears  white. 

395.  Complementary  Colors. — If  certain  colors  of  the 
spectrum  are  combined  in  a  compound  color,  and  the  others  in 
another,  these  two  are  called  complementary  colors,  because,  when 
united,  they  will  produce  white.    For  example,  if  green,  blue,  and 
yellow  are  combined,  they  will  produce  green,  differing  slightly 
from  that  of  the  spectrum;  the  remaining  colors,   red,  orange, 
indigo,  and  violet,  compose  a  kind  of  purple,  unlike  any  color  of 
the  spectrum.     But  these  particular  shades  of  green  and.  purple, 
if  mingled,  will  make  perfectly  white  light,  and  are  therefore 
complementary  colors. 

Tyndal  gives  these  as  complementary  :  Bed  and  greenish  blue, 
orange  and  cyanogen  blue,  yellow  and  indigo  blue,  greenish  yellow 
and  violet. 

396.  Natural  Colors  of  Bodies. — The  colors  which  bodies 
exhibit,  when  seen  in  ordinary  white  light,  are  owing  to  the  fact 
that  they  decompose  light  by  absorbing  or  transmitting  some 
colors  and  reflecting  the  others.     We  say  that  a  body  has  a  certain 
color,  whereas  it  only  reflects  that  color ;  a  flower  is  called  red, 
because  it  reflects  only  or  principally  red  light ;  another  yellow, 
because  it  reflects  yellow  light,  &c.     A  white  surface  is  one  which 
reflects  all  colors  in  their  due  proportion  ;  and  such  a  surface, 
placed  in  the  spectrum,  assumes  each  color  perfectly,  since  it  is 
capable  of  reflecting  all.    A  substance  which  reflects  no  light,  or 
but  very  little,  is  black.     What  peculiarity  of  constitution  that  is 
which  causes  a  substance  to  reflect  a  certain  color,  and  to  absorb 
others  is  unknown. 

Very  few  objects  have  a  color  which  exactly  corresponds  to  any 
color  of  the  spectrum.  This  is  found  to  result  from  the  fact  that 
most  bodies,  while  they  reflect  some  one  color  chiefly,  reflect  the 
others  in  some  degree.  A  red  flower  reflects  the  red  light  abun- 
dantly, and  perhaps  some  rays  of  all  the  other  colors  with  the  red. 
Hence  there  may  be  as  many  shades  of  red  as  there  can  be  differ- 
ent proportions  of  other  colors  intermingled  with  it.  The  same 
is  true  of  each  color  of  the  spectrum.  Thus  there  is  an  infinite 
variety  of  tints  in  natural  objects.  These  facts  are  readily  estab- 
lished by  using  the  prism  to  decompose  the  light  which  bodies 
reflect. 

397.  Fixed  Dark  Lines  of  the  Spectrum. — Let  the  aper- 
ture through  which  the  sunbeam  enters  be  made  exceedingly 
narrow,  and  let  the  prism  be  of  uniform  density,  and  then  let  the 
refracted  pencil  pass  immediately  through  a  small  telescope,  and 
thence  into  the  eye,  and  there  appears  a  phenomenon  of  great 


266 


OPTICS. 


interest—  the  dark  lines,  or  the  Fraunhofer  lines, 
as  they  are  often  called  from  the  name  of  their 
discoverer.  These  lines,  an  imperfect  view  of 
which  is  presented  in  Fig.  254,  are  unequal  in 
breadth,  in  darkness,  and  in  distance  from  each 
other,  and  so  fine  and  crowded  in  many  parts 
that  the  whole  number  cannot  be  counted. 
Fraunhofer  himself  described  between  500  and 
600,  among  which  a  few  of  the  most  prominent 
are  marked  by  letters,  and  used  in  measuring  re- 
fractive power.  At  least  as  many  as  six  thousand 
are  now  known  and  mapped,  so  that  any  one  of 
them  may  be  identified.  They  are  parallel  to 
each  other,  and  perpendicular  to  the  length  of 
the  spectrum.  When  the  pencil  passes  through 
a  succession  of  prisms,  all  bending  it  the  same 
way,  the  spectrum  becomes  more  dilated,  and 
more  lines  are  seen.  The  instrument  fitted  up 
as  above  described,  either  with  one  prism  or  a 
series  of  prisms,  is  called  a  spectroscope. 

398.  Bright  Lines  in  the  Spectrum  of 
Flame.  —  If  the  spectroscope  be  used  for  the 
examination  of  the  flame  of  different  substances 
in  combustion,  the  spectrum  is  found  to  consist 
of  certain  detached  bright  bands,  differing  in 
color  and  number,  according  to  the  substance 
under  examination.  Thus,  the  spectrum  of 
sodium  flame,  besides  showing  other  fainter 
lines,  consists  mainly  of  two  conspicuous  yellow 
lines,  very  close  together,  so  as  ordinarily  to  ap- 
pear as  one.  The  flame  of  carbon  shows  two  dis- 
tinct lines,  one  of  which  is  green,  the  other  indigo. 
In  this  respect  every  substance  differs  from  every 
other,  and  each  may  be  as  readily  distinguished 
by  the  lines  which  compose  its  spectrum  as  by 
any  other  property.  The  lines  of  some  substances 
are  very  numerous  ;  as,  for  example,  iron,  whose 
spectrum  lines  amount  to  four  or  five  hundred. 

But  a  solid  or  liquid  substance,  when  raised 
to  a  red  or  white  heat,  without  passing  into  the 
gaseous  state  and  producing  flame,  forms  a  con- 
tinuous spectrum,  similar  to  that  of  the  sun, 
having  neither  isolated  bright  bands  nor  dark 
lines. 


FIG.  254. 


DISPERSION    OF    LIGHT.  267 

399.  The  Spectrum  of  a  Heated  Solid  or  Liquid  Shin- 
ing through  Flame. — The  condition  of  a  spectrum  is  entirely 
changed  when  the  light  from  a  heated  solid  or  liquid  substance 
shines  through  the  flame  of  a  burning  gas.     The  bright  lines  pecu- 
liar to  that  gas  instantly  become  dark  lines.     The  flame  seems 
to  absorb  just  those  rays,  and  only  those,  which  are  like  the  rays 
emitted  by  itself.     As  an  example,  the  spectrum  of  sodium  flame 
consists  of  a  bright  double  yellow  line,  and  a  few  fine  luminous 
lines  of  other  colors.     If  now  iron  at  an  intense  white  heat 
shines  through  this  flame,   the  whole  spectrum  becomes  lumi- 
nous, except  the  very  lines  which  were  before  bright ;  these  are 
now  dark. 

400.  Composition  of  the  Sun's  Surface. — A  great  num- 
ber of  the  dark  lines  of  the  solar  spectrum  are  identical  in  position 
with  lines  in  the  spectrum  of  terrestrial  substances.     The  spectro- 
scope can  be  attached  to  the  eye-piece  of  a  telescope,  so  as  to  bring 
half  the  breadth  of  the  solar  spectrum  side  by  side  with  half  the 
breadth  of  the  spectrum  of  the  flame  of  some  substance  ;  and  their 
lines  can  thus  be  compared  with  each  other  on  the  divisions  of 
the  same  stjale.    When  this  is  done,  there  is  found,  with  regard  to 
several  substances,  an  identity  of  position  and  relative  breadth  and 
intensity  so  exact  that  it  is  impossible  to  regard  the  agreement  as 
accidental.     The  double  line  D  of  the  sunbeam,  is  the  prominent 
line  of  sodium.     So  all  the  numerous  lines  of  potassium,  iron,  and 
several  other  simple  substances,  exactly  coincide  with  the  dark 
lines  of  the  spectrum  of  sunlight. 

The  foregoing  facts  seem  to  indicate  that  the  photosphere  of 
the  sun  consists  of  the  flame  of  many  substances,  among  which 
are  some  such  as  belong  to  the  earth,  namely,  sodium,  potassium, 
iron,  &c.;  and  that  the  luminous  liquid  matter  beneath  the 
photosphere  shines  through  it,  and  changes  all  the  bright  lines  to 
dark  ones. 

401.  Dispersion  of  Light. — Decomposition  of  light  refers 
to  the  fact  of  a  separation  of  colors  ;  dispersion,  rather  to  the 
measure  or  degree  of  that  separation.     The  dispersive  power  of  a 
medium  indicates  the  amount  of  separation  which  it  produces, 
compared  with  the  amount  of  refraction. 

The  deviation  of  the  line  E  is  usually  taken  as  the  deviation 
of  the  beam  regarded  as  a  whole.  The  difference  of  the  devia- 
tions of  the  lines  A  and  H  is  the  dispersion. 

For  example,  if  a  substance,  in  refracting  a  beam  of  light 
1°  51'  from  its  course,  separates  the  violet  from  the  red  by  4', 
then  its  dispersive  power  is  -fa  =  .036.  The  following  table 


268 


OPTICS. 


gives  the  dispersive  power  of  a  few  substances  much  used  in 
optics : 


Dispersive  power. 

Oil  of  cassia 0.139 

Sulphuret  of  carbon 0. 130 

Oil  of  bitter  almonds 0.079 

Flint-glass 0.052 

Muriatic  acid 0.043 

Diamond 0.038 

Crown-glass 0.036 


Dispersive  power. 

Plate-glass 0.032 

Sulphuric  acid 0.031 

Alcohol 0.029 

Rock-crystal 0.026 

Blue  sapphire  0.026 

Fluor-spar 0.022 


The  discovery  that  different  substances  produce  different  de- 
grees of  dispersion,  is  due  to  Dollond,  who  soon  applied  it  to  the 
removal  of  a  serious  difficulty  in  the  construction  of  optical  in- 
struments. 

402.  Chromatic  Aberration  of  Lenses.— This  is  a  devia- 
tion of  light  from  a  focal  point,  occasioned  by  the  different  re- 
frangibility  of  the  colors.     If  the  surface  of  a  lens  be  covered, 
except  a  narrow  ring  near  the  edge,  and  a  sunbeam  be  transmitted 
through  the  ring,  the  chromatic  aberration  becomes  very  apparent ; 
for  the  most  refrangible  color,  violet,  comes  to  its  focus  nearest, 
and  then  the  other  colors  in  order,  the  focus  of  red  being  most 
remote.     Since  the  distinctness  of  an  image  depends  on  the  ac- 
curate meeting  of  rays  of  the  same  pencil  in  one  point,  it  is  clear 
that  discoloration  and  indistinctness  are  caused  by  the  separation 
of  colors. 

£i 

403.  Achromatism. — In  order  to  refract  light,  and  still  keep 
the  colors  united,  it  is  necessary  that,  after  the  beam  has  been 
refracted,  and  thus  separated,  a  substance  of  greater  dispersive 
power  should  be  used,  which  may  bring  the  colors  together  again, 
by  refracting  the  beam  only  a  part  of  the  distance  back  to  its 
original  direction.      For  instance,   suppose   two  prisms,  one   of 
crown-glass  and  one  of  flint-glass,  each  ground  to  such  a  refract- 
ing angle  as  to  separate  the  violet  from  the  red  ray  by  4'.     In 
order  for  this,  the   crown-glass,  whose  dispersive  power  is  .036, 

4' 

must  refract  the  beam  1°  51' ;  for    0      ,  =  .036  ;  and  the  flmt- 

1    51 

glass,  whose  dispersive  power  is  .052,  must  refract  only  1°  17'; 

4' 
f°r  -10  17;  =  -052.     Place  these  two  prisms  together,  base  to  edge, 

as  in  Fig.  255,  C  being  the  crown-glass  and  F  the  flint-glass. 
Then  0  will  refract  the  beam  b  b.  downward  1°  51',  and  the  violet, 
?%  4'  more  than  the  red,  r ;  F  will  refract  this  decomposed  beam 


ACHROMATIC    LENS. 


269 


upward  1°  17',  and  the  violet  4'  more  than  the  red,  wliich  will  just 
bring  them  together  at  v  r.     Thus  the  colors  are  united  again,  and 

FIG.  255. 


yet  the  beam  is  refracted  downward  1°  51'  —  1°  17'  =  34',  from 
its  original  direction. 

404.  Achromatic  Lens.— If  two  prisms  can  thus  produce 
achromatism,  the  same  may  be  effected  by  lenses ;  for  a  convex 
lens  of  crown-glass  may  converge  the  rays  of  a  pencil,  and  then  a 
concave  lens  of  flint-glass  may  diminish  that  convergency  suffi- 
ciently to  unite  the  colors.  A  lens  thus  constructed  of  two  lenses 
of  different  materials  and  opposite  curvatures,  so  adapted  as  to 
produce  an  image  free  from  chromatic  aberration,  is  called  an 
achromatic  lens.  Fig.  256  shows  such  a  combination.  The  con- 

FIQ.  256. 


vex  lens  of  crown-glass  alone  would  gather  the  rays  into  a  series  of 
colored  foci  from  v  to  r ;  the  concave  flint-glass  lens  refracts  them 
partly  back  again,  and  collects  all  the  colors  at  one  point,  F. 

405.  Colors  not  Dispersed  Proportionally. — It  is  assumed 
in  the  foregoing  discussion,  that  when  the  red  and  violet  are 
united,  all  the  intermediate  colors  will  be  united  also.  It  is  found 
that  this  is  not  strictly  true,  but  that  different  substances  separate 
two  given  colors  of  the  spectrum  by  intervals  which  have  different 
ratios  to  the  whole  length  of  the  spectrum.  This  departure  from 
a  constant  ratio  in  the  distances  of  the  several  colors,  as  dispersed 
by  different  media,  is  called  the  irrationality  of  dispersion.  In 
consequence  of  it  there  will  exist  some  slight  discoloration  in  the 
image,  after  uniting  the  extreme  colors.  It  is  found  better  in 
practice  to  fit  the  curvatures  of  the  lenses,  for  uniting  those  rays 
which  most  powerfully  affect  the  eye. 


270  OPTICS. 

In  a  well-corrected  telescope,  when  pointed  at  a  bright  object, 
such  as  J  upiter  or  the  moon,  a  purple  color  will  be  seen  when  the 
eye-piece  is  pushed  inwards  from  its  position  of  adjustment, 
and  a  greenish  color  will  show  when  the  eye-piece  is  pulled  out 
too  far. 


CHAPTER    V. 

RAINBOW    AND    HALO. 

406,  The   Rainbow. — This  phenomenon,  when  exhibited 
most  perfectly,  consists  of  two  colored  circular  arches,  projected 
on  falling  rain,  on  which  the  sun  is  shining  from  the  opposite  part 
of  the  heavens.     They  are  called  the  inner  or  primary  bow,  and 
the  outer  or  secondary  bow.     Each  contains  all  the  colors  of  the 
spectrum,  arranged  in  contrary  order ;  in  the  primary,  red  is  out- 
ermost ;  in  the  secondary,  violet  is  outermost.    The  primary  bow 
is  narrower  and  brighter  than  the  secondary,  and  when  of  unusual 
brightness,  is  accompanied  by  supernumerary  bows,  as  they  are 
called;  that  is,  narrow  red  arches  just  within  it,  or  overlapping 
the  violet ;  sometimes  three  or  four  supernumeraries  can  be  traced 
for  a  short  distance.    The  common  centre  of  the  bows  is  in  a  line 
drawn  from  the  sun  through  the  eye  of  the  spectator. 

407.  Action  of  a  Transparent  Sphere  on  Light. — Let 

a  hollow  sphere  of  glass  be  filled  with  water,  and  cause  a  beam  of 
parallel  rays  of  homogeneous  yellow  light  to  fall  upon  it.  To  pre- 
vent confusion  in  Fig.  257,  we  will 
consider  only  those  rays  which  fall 
upon  the  upper  half  of  the  section 
of  the  sphere,  and  will  trace  them 
as  they  emerge  at  the  lower  half. 
Those  rays  which  enter  near  the 
axis  8'  m  will  be  refracted  to  points 
near  n.  Rays  still  farther  from 
8'  m  will  be  refracted  to  points  still 
farther  from  n.  Rays  at  about  59° 
from  m,  at  A,  will  be  refracted  to 
J5,  and  no  ray,  no  matter  where  it 
may  enter  the  sphere,  can  be  re- 
fracted to  a  point  higher  than  B.  Now  as  B  is  the  limit  of  the 
arc  n  Bt  it  follows  that  rays  close  to  the  middle  ray  of  the  pencil 


THE    PRIMARY    BOW.  271 

£  Ay  both  above  it  and  below  it,  will  be  refracted  to  B,  crowded 
together  as  it  were,  and  after  reflection  a  large  portion  will  emerge 
at  D.  As  on  passing  into  the  sphere  at  A  from  air,  these  converged 
to  B,  so  on  emerging  the  reverse  action  takes  place  at  />,  and  we 
have  a  compact  pencil  of  parallel  rays.  An  eye  at  E  would  receive 
an  impression  of  bright  light  in  the  direction  E  D  ;  an  eye  below 
ED  would  receive  no  light  at  all,  and  at  E'  or  E",  while  some 
light  would  be  received  from  the  diverging  rays,  the  impression 
would  be  much  less  vivid  than  at  E.  "We  have  been  considering 
only  a  section  of  the  sphere  through  the  axis  ;  if  now  we  conceive 
this  section  to  revolve  about  the  axis  S'  C,  our  beam  S  A  becomes 
a  hollow  cylinder  of  light,  and  the  emergent  beam  E  D  becomes 
an  emergent  hollow  frustum  of  a  cone,  and  if  the  eye  be  placed 
at  any  element  of  this  cone  the  effect  will  be  as  described  for  the 
element  E. 

408.  The  Primary  Bow. — In  the  preceding  article,  homo- 
geneous yellow  light  was  considered.     Let  us  examine  the  results 
when  white  light  from  the 

sun  falls  upon  a  rain-drop.  FIG.  258. 

Suppose  8  A  (Fig.  258)  to 
be  a  beam  of  parallel  rays 
from  the  sun  incident  at 
59°  from  m.  As  B  was  the 
point  at  which  yellow  rays 
of  the  beam  were  concen- 
trated, the  red  rays  which 
are  less  refrangible  will 
all  concentrate  at  R,  the 
distance  R  B  being  very 
greatly  exaggerated  for  the 
sake  of  clearness  in  the  dia- 
gram. After  reflection  these  red  rays  will  emerge  as  a  beam  of 
parallel  red  rays  at  R'.  The  violet  rays  of  the  beam  S  A  being 
most  refrangible  will  all  meet  at  V,  below  B,  and  will  emerge  at 
V  as  a  beam  of  parallel  violet  rays.  Between  these  will  be  beams 
of  the  intermediate  colors  of  the  spectrum.  These  are  beams  of 
parallel  rays,  but  are  not  parallel  beams,  as  is  shown  in  the  figure. 
The  angle  #  included  between  the  incident  beam  8  A  and  the 
emergent  red  beam  produced  backward  to  x  is  found  by  cal- 
culation to  be  42°  2',  and  the  like  angle  for  the  violet  beam  is 
40°  17'. 

409.  Course   of  Rays  in  Secondary  Bow.— If  we  ex- 
amine the  conditions  of  two  internal  reflections  (Fig.  259),  wo 
find  that  a  beam  of  monochromatic  light  entering  at  a  certain 


272 


OPTICS. 


FIG.  259. 


distance  from  the  axis  8'  n,  about  71°  42',  suffers  the  least  devia- 
tion possible  after  two  reflections,  that  is  to  say,  the  angle  8x1 

will  be  a  minimum.  Rays 
near  this  ray  8  A  of  mini- 
mum deviation  both  on 
the  side  towards  the  axis 
8 '  n  and  on  the  side  away 
from  it  will  tend  to  meet 
in  a  focus  at  /  about  f  of 
the  distance  A  B,  and  will 
then  be  reflected  parallel 
to  Z>,  again  being  reflected 
to  a  focus  at/'  (E  f  =  £ 
E  Z>),  and  finally  emerg- 
ing at  E,  a  parallel  beam  as  on  entering.  An  observer  at  / 
would  receive  an  intense  beam  of  light  of  the  particular  color 
used. 

Now  substitute  for  the  monochromatic  light,  light  from  the 
sun,  and  the  results  will  be  as  illustrated  in  Fig.  260,  in  which 
the  difference  in  direction  between  the  red  and  the  violet  rays  has 
been  greatly  exaggerated.  At  A  the  red  rays,  following  the  course 
given  in  Fig.  260  are  con- 


verged, cross  at  the  focus, 
and  at  r  are  reflected  as  a 
parallel  beam  to  rf ;  here 
they  are  again  reflected  to 
a  focus,  and  again  diverg- 


FIG.  260. 


ing  pass 


on   to  r",  where 


they  emerge  as  a  parallel 
beam  r"  JR.  The  violet 
rays  are  separated  from 
the  red  at  A,  and  being 
more  refrangible,  take  the  path  indicated  in  the  figure.  The 
angle  A  x  r"  =  50°  59'  and  A  x'  v"  =  549  9'.  In  order  that  the 
emergent  pencil  may  enter  the  eye  the  incident  ray  must  enter  on 
the  side  of  the  drop  nearest  the  observer.  The  rays  just  con- 
sidered are  the  only  ones  which,  after  two  reflections,  emerge  com- 
pact and  parallel,  and  give  bright  color  at  a  great  distance. 

The  explanation  just  concluded  gives  a  general  and  sufficiently 
exact  conception  of  the  phenomena  of  the  primary  and  secondary 
bows. 

A  rigid  mathematical  analysis  would  take  note  of  the  caustics 
by  reflection  and  refraction,  and  would  vary  slightly  the  places  of 
maximum  illumination.  To  such  analysis,  and  to  the  principle 
of  interference,  the  student  must  refer  for  an  account  of  the  super- 


AXIS    OF    THE    BOWS. 


273 


numerary  bows  accompanying  the  primary,  which  are  sometimes 
observed. 

410.  Axis  of  the  Bows.— Let  A  B  D  G  I  (Fig.  261)  repre- 
sent the  path  of  the  pencil  of  red  light  in  the  primary  bow.  If 
A  B  and  /  G  are  produced  to  meet  in  K,  the  angle  K  is  the  devi- 
ation, 42°  2',  of  the  incident  and  emergent  red  rays.  Suppose  the 
spectator  at  /,  and  let  a  line  from  the  sun  be  drawn  through  his 
position  to  T\  it  is  sensibly  parallel  to  A  B,  and  therefore  the 
ingles  /  and  K  are  equal.  As  T  is  opposite  to  the  sun,  the  red 

FIG.  262. 


color  is  seen  at  the  distance  of  42°  2',  on  the  sky,  from  the  point  T: 
and  so  the  angular  distance  of  each  color  from  T  equals  the  angle 
which  the  ray  of  that  color  makes  with  the  incident  ray.  In  like 
manner,  in  the  secondary  bow,  HIT  (Fig.  262)  be  drawn  through 
the  sun  and  the  eye  of  the  observer,  it  is  parallel  to  A  B,  and  the 
angular  distance  of  the  colored  ray  from  Tis  equal  to  K,  the  devi- 
ation of  the  incident  and  emergent  rays.  /  T  is  called  the  axis  of 
the  bows,  for  a  reason  which  is  explained  in  the  next  article. 

411.  Circular  Form  of  the  Bows.— Let  8  0  C  (Fig.  263) 
be  a  straight  line  passing  from  the  sun,  through  the  observer's 

FIG.  263. 


place  at  0,  to  the  opposite  point  of  the  sky ;  and  let  V  0,  R  0  be 
the  extreme  rays,  which  after  one  reflection  bring  colors  to  the  eye 
18 


274  OPTICS. 

at  0,  and  R'  0,  V  0,  those  which  exhibit  colors  after  two  reflec- 
tions ;  then  (according  to  Arts.  408,  409),  V  0  C  =  40°  17',  R  0  C 
=  42°  2',  R'  0  0=  50°  59',  V  0  C=  54°  9'.  Now,  if  we  sup- 
pose the  whole  system  of  lines,  S  V '  0,  8  V  0,  to  revolve  about 
S  0  C,  as  an  axis,  the  relations  of  the  rays  to  the  drops,  and  to 
each  other,  will  not  be  at  all  changed  ;  and  the  same  colors  will 
describe  the  same  lines,  whatever  positions  those  lines  may  occupy 
in  the  revolution.  The  emergent  rays,  therefore,  all  describe  the 
surfaces  of  cones,  whose  common  vertex  is  in  the  eye  at  0  ;  and 
the  colors,  as  seen  on  the  cloud,  are  the  circumferences  of  their 
bases. 

In  a  given  position  of  the  observer,  the  extent  of  the  arches 
depends  on  the  elevation  of  the  sun.  When  on  the  horizon,  the 
bows  are  semicircles;  but  less  as  the  sun  is  higher,  because  their 
centre  is  depressed  as  much  below  the  horizon  as  the  sun  is  ele- 
vated above  it.  From  the  top  of  a  mountain,  the  bows  have  been 
seen  as  almost  entire  circles. 

412.  Colors  of  the  Two  Bows  in  Reversed  Order. — 

Suppose  the  eye  to  receive  a  red  ray  from  a  drop  a  (Fig.  2G4) ; 

rays  of  all  other  colors  being  more  refrangible  than  the  red  would 
above  the  eye,  as  does  v'.  In  order  that  a  violet  ray  may 
enter  the  eye  it  must  proceed  from  a 
lower  drop,  as  b,  and  the  less  refrangible 
rays  from  this  drop  will  pass  below  the 
eye,  as  at  R'.  Hence  in  the  primary  bow 
the  drops  which  send  violet  to  the  eye 
are  nearer  to  the  axis  of  the  bow  than 
those  which  send  red,  red  being  there- 
fore the  outermost  color.  A  like  exami- 
nation of  the  secondary  bow  shows  that 

red  is  the  innermost  and  violet  the  outermost  color. 

413.  Rainbows,  the  Colored  Borders  of  Illuminated 
Segments  of  the  Sky. — The  primary  bow  is  to  be  regarded  as 
the  outer  edge  of  that  part  of  the  sky  from  which  rays  can  come  to 
the  eye  after  suffering  but  one  reflection  in  drops  of  rain  ;  and  the 
secondary  bow  is  the  inner  edge  of  that  part  from  which  light, 
after  being  twice  reflected,  can  reach  the  eye. 

It  is  found  by  calculation,  that  in  case  of  one  reflection,  the 
incident  and  emergent  rays  can  make  no  inclinations  with  each 
other  greater  than  42°  2'  for  red  light,  and  40°  1?'  for  violet ;  but 
the  inclinations  may  be  less  in  any  degree  down  to  0°.  There- 
fore, all  light,  once  reflected,  comes  to  the  eye  from  within  the 
primary  bow. 

Bub  the  angles,  50°  59'  and  54°  9',  are,  by  calculation,  the  least 


THE    COMMON    HALO. 


275 


deviations  of  red  and  violet  light  from  the  incident  rays  after  two 
reflections.  But  the  deviations  may  be  greater  than  these  limits 
up  to  180°.  Therefore  rays  twice  reflected  can  come  to  the  eye 
from  any  part  of  the  sky,  except  between  the  secondary  bow  and 
its  centre. 

It  appears,  then,  that  from  the  zone  lying  between  the  two 
bows,  no  light,  reflected  by  drops  internally,  either  once  or  twice, 
can  possibly  reach  the  eye.  Observation  confirms  these  state- 
ments ;  when  the  bows  are  bright,  the  rain  within  the  primary  is 
more  luminous  than  elsewhere ;  and  outside  of  the  secondary  bow, 
there  is  more  illumination  than  between  the  two  bows,  where  the 
cloud  is  perceptibly  darkest. 

414.  The  Common  Halo.— This,  as  usually  seen,  is  a  white 
or  colored  circle  of  about  22°  radius,  formed  around  the  sun  or 
moon.     It  might,  without  impropriety,  be  termed  the  frost-bow, 
since  it  is -known  to  be  formed  by  light  refracted  by  crystals  of  ice 
suspended  in  the  air.     It  is  formed  when  the  sun  or  moon  shines 
through  an  atmosphere  somewhat  hazy.    About  the  sun  it  is  a 
white  ring,  with  its  inner  edge  red,  and  somewhat  sharply  defined, 
while  its  outer  edge  is  colorless,  and  gradually  shades  off  into  the 
light  of  the  sky.     Around  the  moon  it  differs  only  in  showing 
little  or  no  color  on  the  inner  edge. 

415.  How  Caused. — The  phenomenon  is  produced  by  light 
passing  through  crystals  of  ice,  having  sides  inclined  to  each  other 
at  an  angle  of  60°.     Let  the  eye  be  at  E  (Fig.  265),  and  the  sun  in 
the  direction  E  8.    Let  8  A,  SB,  &c.,  be  rays  striking  upon  such 
crystals  as  may  happen  to  lie  in  a  position 

to  refract  the  light  toward  S  E  as  an  axis. 
Each  crystal  turns  the  ray  from  the  refract- 
ing edge  on  entering ;  and  again,  on  leav- 
ing, it  is  bent  still  more,  and  the  emergent 
pencil  is  decomposed.  The  color,  which 
comes  from  each  one  to  the  eye  E,  depends 
on  its  angular  distance  from  E  S,  and 
the  position  of  its  refracting  angle.  The 
angle  of  deviation  for  ^4  is  EA  D— SEA  ; 
for  B,  it  is  S  E  B,  and  so  on.  It  is  found 
by  calculation,  that  the  least  deviation  for 
red  light  is  21°  45' ;  the  least  for  orange 
must  be  a  little  greater,  because  it  is  a 
little  more  refrangible,  and  so  on  for  the 
colors  in  order.  The  greatest  deviation 
for  the  rays  generally  is  about  43°  13'.  All 
light,  therefore,  which  can  be  transmitted 


FIG.  265. 


276  OPTICS. 

by  such  crystals  must  come  to  the  observer  from  points  some- 
where between  these  two  limits,  21°  45'  and  43°  13'  from  the  sun. 
But  by  far  the  greater  part  of  it,  as  ascertained  by  calculation, 
passes  through  near  the  least  limit. 

416.  Its  Circular  Form. — What  takes  place  on  one  side  of 
E  8  may  occur  on  every  side  ;  or,  in  other  words,  we  may  sup- 
pose the  figure  revolved  about  E  8  as  an  axis,  and  then  the  trans- 
mitted light  will  appear  in  a  ring  about  the  sun  8.     The  inner 
edge  of  the  ring  is  red,  since  that  color  deviates  least ;  just  out- 
side of  the  red  the  orange  mingles  with  it ;  beyond  that  are  the 
red,  orange,  and  yellow  combined ;  and  so  on,  till,  at  the  minimum 
angle  for  violet,  all  the  colors  will  exist  (though  not  in  equal  pro- 
portions), and  the  violet  will  be  scarcely  distinguishable  from 
white.    Beyond  this  narrow  colored  band  the  halo  is  white,  grow- 
ing more  and  more  faint,  so  that  its  outer  limit  is  not  discernible 
at  all. 

417.  The  Halo,  a  Bright  Border  of  an  Illuminated 
Zone. — As  in  the  rainbow,  so  in  the  halo,  the  visible  band  of 
colors  is  only  the  border  of  a  large  illuminated  space  on  the  sky. 
The  ordinary  halo,  therefore,  is  the  bright  inner  border  of  a  zone, 
which  is  more  than  20°  wide.     The  whole  zone,  except  the  inner 
edge,  is  too  faint  to  be  generally  noticed,  though  it  is  perceptibly 
more  luminous  than  the  space  between  the  halo  and  the  luminary. 

418.  Frequency  of  the  Halo. — The  halo  is  less  brilliant 
and  beautiful,  but  far  more  frequent,  than  the  rainbow.     Scarcely 
a  week  passes  during  the  whole  year  in  which  the  phenomenon 
does  not  occur.     In  summer  the  crystals  are  three  or  four  miles 
high,  above  the  limit  of  perpetual  frost.     As  the  rainbow  is  some- 
times seen  in  dew-drops  on  the  ground,  so  the  frost-bow,  just  after 
sunrise,  has  been  noticed  in  the  crystals  which  fringe  the  grass. 

419.  The  Mock  Sun. — The  mock  sun,    or  sun-dog,  is  a 
short  arc  of  the  halo,  occasionally  seen  at  22°  distance,  on  the 
right  and  left  of  the  sun,  when  near  the  horizon.     The  crystals, 
which  are  concerned  in  producing  the  mock  sun,  are  supposed  to 
have  the  form  of  spiculce,  or  six-sided  needles,  whose  alternate 
sides  are  inclined  to  each  other  at  an  angle  of  60°  ;  these  falling 
through  the  air  with  their  axes  vertical,  refract  the  light  only  in 
directions  nearly  horizontal,  and  therefore  present  only  the  right 
and  left  sides  of  the  halo. 

In  high  latitudes,  other  and  complex  forms  of  halo  are  fre- 
quent, depending  for  their  formation  on  the  prevalence  of  crys- 
tals of  other  angles  than  60°.  [See  Appendix  for  calculations  of 
the  angular  radius  of  rainbows  and  halo.] 


THE     WAVE    THEORY 


277 


CHAPTER    VI. 

NATURE    OF    LIGHT.— WAVE    THEORY. 

420.  The  Wave  Theory. — Light  has  sometimes  been  re- 
garded as  consisting  of  material  particles  emanating  from  lumi- 
nous bodies.     But  this,  called  the  corpuscular  or  emission  theory, 
has  mostly  yielded  to  the  undulatory  or  wave  theory,  which  sup- 
poses light  to  consist  of  vibrations  in  a  medium.     This  medium, 
called  the  luminiferous  ether,  is  imagined  to  exist  throughout  all 
space,  and  to  be  of  such  rarity  as  to  pervade  all  other  matter. 
It  is  supposed  also  to  be  elastic  in  a  very  high  degree,  so  that 
undulations  excited  in  it  are  transmitted  with  great  velocity. 

There  is  no  independent  evidence  of  the  existence  of  this 
theoretical  ether. 

Kadiant  heat  consists  of  undulations  of  the  same  ether,  which 
differ  from  those  of  light  only  in  being  slower.  For  it  is  a 
familiar  fact,  that  when  the  heat  of  a  body  is  increased  to  about 
500°  or  600°  C.  the  body  becomes  luminous,  and  the  brightness 
increases  as  the  temperature  is  raised. 

421.  Nature  of  the  Wave. — Suppose  a  number  of  ether 
molecules,  as  a,  b,  c,  &c.  (Fig.  266),  to  be  equidistant  upon  the 
straight  line  a  f.     Now  conceive 

that  a  moves  to  a',  then  back  to 
a",  thence  to  a  again,  occupying 
four  equal  intervals  of  time  in 
the  circuit.  Suppose  b  to  start 
on  the  same  round,  at  a  time  one 
interval  later  ;  when  a  reaches 
its  original  position,  and  is  just  beginning  its  upward  motion,  as 
in  the  figure,  b  will  be  at  b"  moving  towards  b.  In  like  manner, 
starting  c  and  d  at  intervals  later  by  one,  we  shall  find  their  posi- 
tions and  directions  of  motion,  when  a  begins  its  second  circuit, 
to  be  as  given  in  the  figure.  The  motion  will  have  been  trans- 
mitted to  e,  which  will  begin  its  first  circuit  at  the  instant  that  a 
starts  upon  its  second.  Molecules  which  like  a  and  e  are  in  the 
same  condition  as  to  place  and  direction  of  motion,  are  said  to  be 
in  the  same  phase.  A  wave  length  is  the  distance  between 
two  consecutive  like  phases.  The  amplitude  of  vibration  is  the 
distance  between  the  two  limits  of  the  excursion  of  the  particle. 
It  is  evident  from  the  figure  that  the  motion  r<  communicated 


278  OPTICS. 

along  the  axis  fl/a  distance  a  ey  or  one  wave  length,  during  the 
of  one  vibration. 


422.  Postulates  of  the  Wave  Theory.— 

1.  The  waves  are  propagated  through  the  ether  at  the  rate  of 
186,800  miles  per  second. 

As  this  is  the  known  velocity  of  light,  it  must  be  the  rate  at 
which  the  waves  are  transmitted. 

2.  The  atoms  of  the  ether  vibrate  at  right  angles  to  the  line  of 
the  ray  in  all  possible  directions. 

It  was  at  first  assumed  that  the  luminous  vibrations,  like  the 
vibrations  of  sound,  are  longitudinal,  that  is,  back  and  forth  in  the 
line  of  the  ray  ;  but  the  discoveries  in  polarization  require  that  the 
vibrations  of  light  should  be  assumed  to  be  transverse,  that  is,  in 
a  plane  perpendicular  to  the  line  of  the  ray  ;  and,  moreover,  that 
in  that  plane  the  vibrations  are  in  every  possible  direction  within 
an  inconceivably  short  space  of  time.  Thus,  if  a  person  is  look- 
ing at  a  star  in  the  zenith,  we  must  consider  each  atom  of  the 
ether  between  the  star  and  his  eye  as  vibrating  across  the  vertical 
in  all  horizontal  directions,  north  and  south,  east  and  west,  and 
in  innumerable  lines  between  these. 

3.  Different  colors  are  caused  by  different  rates  of  vibration. 
Red  is  caused  by  the  sloivest  vibrations,  and  violet  by  the 

quickest,  and  other  colors  by  intermediate  rates.  White  light  is 
to  the  eye  what  harmony  is  to  the  ear,  the  resultant  effect  of 
several  rates  of  vibration  combined.  There  are  slower  vibrations 
of  the  ether  than  those  of  red  light,  and  quicker  ones  than  those 
of  violet  light,  but  they  are  not  adapted  to  affect  the  vision.  The 
former  affect  the  sense  of  feeling  as  heat,  the  latter  produce 
chemical  effects,  and  are  called  actinic  rays. 

4.  The  ether  ivithin  bodies  is  less  elastic  than  in  free  space. 
This  is  inferred  from  the  fact  that  light  moves  with  less  veloc- 

ity in  passing  through  bodies  than  in  free  space  ;  the  greater  the 
refractive  power  of  a  body,  the  slower  does  light  move  within  it. 
And  in  some  bodies  of  crystalline  structure,  it  happens  that  the 
velocity  is  different  in  different  directions,  so  that  the  elasticity  of 
the  ether  within  them  must  be  regarded  as  varying  with  the 
direction. 

423.  Reflection  and  Refraction  according  to  the  Wave 
Theory.  —  The  vibrations  of  the  ether  are  transmitted  from  the 
source  of  motion  as  spherical  waves.     In  a  luminous  body  are  an 
infinite  number  of  radiants,  each  the  centre  of  a  succession  of 
spherical  waves.    A  beam  of  parallel  rays  is  a  collection  of  parallel 
radii  of  spherical  waves,  having  different  centres  of  disturbance, 
and  the  wave  front  of  such  a  beam  is  the  tangent  plane  common 
to  all  the  spheres. 


RATIO    OP    ANGLES. 


279 


FIG.  267. 


Suppose  A  and  B  to  be  two  parallel  rays  of  a  beam  of  light. 
Let  a  and  b  (Fig.  267)  represent  two  like  wave  fronts,  and  a  a'  = 
b  V  be  the  distance  light  moves 
in  any  small  unit  of  time.  When 
the  wave  a  reaches  #',  b  will  have 
reached  b'.  While  b'  moves  to 
b",  a'  regarded  as  a  centre  of  dis- 
turbance will  have  sent  out  a 
spherical  wave  to  a".  While  b" 
is  transmitting  a  spherical  wave 
to  b",  a"  will  have  extended  to 
a'",  and  the  common  tangent 
plane  b'"  a'"  will  be  the  wave 
front,  and  A'  Br,  the  reflected 
rays,  represent  the  beam.  All  rays  from  a  and  b"  which  move 
obliquely  with  respect  to  each  other  are  destroyed  by  their  mutual 
reactions,  only  those  remaining  which  move  in  parallel  directions, 
as  A'  B'. 

While  b'  moves  to  b",  a'  is  sending  a  wave  into  the  medium  at 
a  slower  rate,  suppose  with  only  half  the  velocity,  which  has 
advanced  to  c  by  the  time  b'  reaches  b".  While  b"  is  moving  into 
the  medium  to  d,  c  has  moved  to  cf,  and  a  common  tangent  d  c'  is 
the  front  of  the  refracted  beam,  of  which  A"  B"  are  rays. 

424.  Relation  of  Angles  of  Incidence,  Reflection,  and 
Refraction. — The  velocity  of  light  in  any  medium  being  uniform, 

retaining  the  notation  of  Fig. 
267  we  have  in  the  triangles 
a'  b'  b"  and  a'  a"  b"  (Fig.  268) 
a'  a"  =  b'  b",  a'  b"  common, 
and  the  angles  at  a"  and  b' 
equal,  being  right  angles 
formed  by  the  radii  and  tan- 
gents ;  hence  the  angles  a"  a'  b" 
and  b'  b"  a'  are  equal ;  there- 
fore the  incident  and  reflected  rays  must  make  equal  angles  with 
the  surface,  and  consequently  with  the  normal. 

In  the  triangle  a'  b'  b"  we  have  V  b"  :  a'  b"  : :  sine  b'  a'  b"  : 
sine  90° ;  and  from  a'  b"  c  we  have 

a'  c  i  a'  b"  : :  sine  a'  b"  c  :  sine  90° ; 

combining  these,  b'  b"  :  a'  c  ::  sine  b'  a'  b"  :  sine  a'  b"  c ;  or, 
b'  b"  _  sine  b'  a1  b"  _  sine  x  a'  a  _      sine  ang.  Inc. 
a'  c   '~  sine  a'  b"  c    '  sine  c  a'  y  ~~  sine  ang.  Refrac.' 
But  b'  b"  and  a'  c  are  spaces  through  which  the  wave  is  propa- 


FIG.  268. 


280  OPTICS. 

gated  in  the  same  interval  of  time,  and  as  the  velocities  are  con- 

-It       Til 

stant  in  each  medium,  the  ratio  -,  —  must  be  a  constant  ratio  ; 

,,       »       .,  ,      .  .       sine  ang.  Inc. 

therefore  its  equal  ratio  must  also  be  a  constant 


ratio. 

In  the  above  it  has  been  assumed  that  the  velocity  of  trans- 
mission in  the  denser  medium  is  less  than  in  the  rarer;  this 
assumption  is  verified  by  direct  experiment. 

425.  Interference.  —  As  two  systems  of  water-waves  may 
increase  or  diminish  their  height  by  being  combined,  and  as 
sounds,  when  blended,  may  produce  various  results,  and  even 
destroy  each  other,  so  may  two  pencils  of  light  either  augment  or 
diminish  each  other's  brightness,  and  even  produce  darkness. 

If  unlike  phases  coincide,  the  vibrations  are  destroyed  and 
darkness  follows,  while  if  like  phases  meet,  increase  of  brightness 
results. 

To  illustrate  this,  let  two  plane  reflectors,  inclined  at  a  very 
obtuse  angle,  nearly  180°,  receive  light  from  a  minute  radiant,  and 
reflect  it  to  one  spot  on  a  screen  ;  the  reflected  pencils  will  inter- 
fere, and  produce  bright  and  dark  lines.  Suppose  light  of  one 
color,  as  violet,  flows  from  a  radiant  point  A  (Fig.  269)  ;  let  mir- 
rors B  C  and  B  D  reflect  it  to  the  screen  K  L.  F  and  E  may  be 
so  selected  that  the  ray  A  F  -f  F  G  equals  the  ray  A  E  -\-  E  G. 

FIG.  269. 


Then  G  will  be  luminous,  because  the  two  paths  being  equal,  the 
same  phase  of  wave  in  each  ray  will  occur  at  the  point  G.  But  if 
H  be  so  situated  that  A  f  +  /  H  differs  half  a  violet  ivave  from 
A  e  +  e  H,  then  H  will  be  a  dark  point,  because  opposite  phases 
meet  there.  A  similar  point,  /,  will  lie  on  the  other  side  of  G. 
Again,  there  are  two  points,  K  and  L,  one  on  each  side  of  G,  to 
each  of  which  the  whole  path  of  light  by  one  mirror  will  exceed 
the  whole  by  the  other  by  just  one  violet  wave ;  those  points  are 
bright. 

If  the  paths  differ  by  any  odd  multiple  of  £  wave  length,  light 
is  destroyed  and  a  dark  band  is  seen  ;  but  if  these  paths  differ  by 


STRIATED    SURFACES.  281 

any  multiple  of  a  whole  wave  length,  the  light  is  intensified  and 
bright  bands  are  seen. 

Thus,  there  is  a  series  of  bright  and  dark  points  on  the 
screen  ;  or  rather  a  series  of  bright  and  dark  hyperbolic  lines,  of 
which  these  points  are  sections.  Other  colors  will  give  bands 
separated  a  little  further,  indicating  longer  waves.  And  white 
light,  producing  all  these  results  at  once,  will  give  a  repetition  of 
the  prismatic  series. 

426.  Striated  Surfaces. — If  the  surface  of  any  substance  is 
ruled  with  fine  parallel  grooves,  2000  or  more  to  the  inch,  these 
grooves  will  act  like  the  inclined  mirrors  of  the  last  paragraph, 
and  it  will  reflect  bright  colors  when  placed  in  the  sunbeam. 
Mother-of-pearl  and  many  kinds  of  sea-shell  exhibit  colors  on 
account  of  delicate  striae  on  their  surface.     It  may  be  known  that 
the  color  arises  from  such  a  cause,  if,   when  the  substance  is 
impressed  on  fine  cement,  its  colors  are  communicated  to  the 
cement.     Indeed,  it  was  in  this  way  that  Dr.  Wollaston  accident- 
ally discovered  the  true  cause  of  such  colors.     The  changeable 
hues  in  the  plumage  of  some  birds,  and  the  wings  of  some  insects, 
are  owing  to  a  striated  structure  of  their  surfaces.    But  the  metals 
can  be  made  to  furnish  the  most  brilliant  spectra,  by  stamping 
them  with  steel  dies,  which  have  been  first  ruled  by  a  diamond 
with  lines  from  2000  to  10,000  per  inch,  and  then  hardened. 
Gilt  buttons  and  other  articles  for  dress  are  sometimes  prepared 
in  this  manner,  and  are  called  iris  ornaments.     The  color  in  a 
given  case  depends  on  the  distance  between  the  grooves,  and  the 
obliquity  of  the  beam  of  light.     Hence,  the  same  surface,  uni- 
formly striated,  may  reflect  all  the  colors,  and  every  color  many 
times,  by  a  mere  change  in  its  inclination  to  the  beam  of  light. 

427.  Thin    Laminae. — Any  transparent  substance,    when 
reduced  in  thickness  to  a  few  millionths  of  an  inch,  reflects 

brilliant  colors,  which   vary  with  every 
FIG.  270.  change  of  thickness.     Examples  are  seen 

in  the  thin  laminae  of  air  occupying 
cracks  in  glass  and  ice,  and  the  inter- 
stices between  plates  of  mica,  also  in  thin 
films  of  oil  on  water,  and  alcohol  on  glass, 
but  most  remarkably  in  soapy  water  blown 
into  very  thin  bubbles. 

Let  a  beam  8  o  of  red  light  (Fig.  270) 

be  incident  upon  the  first  surface  A  B  of  the  thin  plate ;  a  part 
will  be  reflected  in  o  m  while  another  part  will  be  refracted  to  r 
in  the  second  surface  A'  B ';  this  refracted  beam  will  again  divide, 
a  portion  being  reflected  to  o'  and  then  refracted  in  o'  m'  •  o  m 


282  OPTICS. 

and  o'  m'  are  parts  of  the  same  beam  and  will  reinforce  each 
other,  or  will  interfere  and  destroy  each  other,  according  to  the 
phases  which  are  superimposed.  At  each  internal  reflection, 
owing  to  the  change  in  density  and  elasticity  at  the  surfaces  of 
contact  of  the  medium  and  air,  there  is  a  loss  or  retardation  equal 
to  one  half  a  wave  length. 

Calling  the  part  of  the  path  o  r  o,  included  between  the  sur- 
faces, 2  t,  then  whenever  2  t  is  a  multiple,  x  I,  of  a  whole  wave 
length,  the  difference  of  the  paths  8  o  m  and  8  o  r  o'  m'  will  be 

x  I  -f-  -,  -  being  the  retardation  spoken  of  above,  and  the  inter- 
Z    Z 

ference  results  in  a  dark  band.    When  2  t  is  an  odd  multiple  of 

3  I  31       I 

one-half  a  wave  length,  suppose  — ,  then  --  +  -  =2  Z,  and  there- 

z  z      z 

fore  o  m  and  o'  m'  will  reinforce  each  other,  producing  a  light 
band. 

If  we  examine  the  transmitted  rays  8  n  and  S  r  o'  ri,  the 
latter  having  been  twice  reflected  internally,  we  find  the  difference 
of  path  to  be  2  t,  and  as  two  half-wave  lengths  must  be  added  to 
this  because  of  the  two  internal  reflections  we  have  for  the  retar- 
dation of  r'  ri,  as  compared  with  r  n,  2  t  -f-  I.  When  2  t  is  a 
multiple  of  a  whole  wave  length,  there  will  be  no  interference, 
and  when  2  t  is  an  odd  multiple  of  one-half  a  wave  length  there 
will  be  interference,  results  the  opposite  of  those  found  for 
reflected  light;  hence  when  a  thin  plate  shows  no  light  by  reflec- 
tion it  shows  bright  light  by  transmission. 

If  we  use  a  film  of  air  of  varying  thickness,  and  the  light  of 
the  sun,  various  colors  will  be  produced. 

If  a  lens  of  slight  convexity  is  laid  on  a  plane  lens,  and  the 
two  are  pressed  together  by  a  screw,  and  viewed  by  reflected  light, 
rings  of  color  are  seen  arranged  around  the  point  of  contact.  The 
rings  of  least  diameter  are  broadest  and  most  brilliant,  and  each 
one  contains  the  colors  of  the  spectrum  in  their  order,  from  violet 
on  the  inner  edge  to  red  on  the  outer.  But  the  larger  rings  not 
only  become  narrower  and  paler,  but  contain  fewer  colors;  yet  the 
succession  is  always  in  the  same  order  as  above.  Increased  pres- 
sure causes  the  rings  to  dilate,  while  new  ones  start  up  at  the 
centre,  and  enlarge  also,  until  the  centre  becomes  black,  after 
which  no  new  rings  are  formed.  These  are  commonly  called 
Newton's  rings,  because  Sir  Isaac  Newton  first  investigated  their 
phenomena. 

428.  Ratio  of  Thicknesses  for  Successive  Rings. — A 

given  color  appears  in  a  circle  around  the  point  of  contact,  because 
equal  thicknesses  are  thus  arranged.  If  the  diameters  of  the  sue- 


NEWTON'S    RINGS.  283 

cessive  rings  of  any  one  color  be  carefully  measured,  their  squares 
are  found  to  be  as  the  odd  numbers,  1,  3,  5,  7  ;  and  hence  the 
thicknesses  of  the  laminae  of  air  at  the  repetitions  of  the  same 
color  are  as  the  same  numbers.  For,  let  Fig.  271  represent  a  sec- 
tion of  the  spherical  and  plane 
surfaces  in  contact  at  a.  Let  IG' 

a  b,  a  d,  be  the  radii  of  two 
rings  at  their  brightest  points. 
Suppose  a  i,  perpendicular  to 
m  n,  to  be  produced  till  it 
meets  the  opposite  point  of 

the  circle  of  which  a  g  is  an  arc,  and  call  that  point/;  then  a  /is 
the  diameter  of  the  sphere  of  which  the  lens  is  a  segment.     Let 
b  e,  d  g,  be  parallel  to  a  i,  and  e  h,  g  i,  to  m  n,  then  we  have 
(e  h)*  :  (g  if  :  :  a  h  x  hf  :  a  i  x  if. 

But  the  distances  between  the  two  lenses  being  exceedingly 
email  in  comparison  with  the  diameter  of  the  sphere,  h  /and  i  f 
may  be  taken  as  equal  to  #/;  whence,  by  substitution, 

(e  h)2  :  (g  i)*  :  :  a  h  x  af:ai  x  af::ah:ai::be:dg. 
Therefore  the  thicknesses  of  successive  rings  are  as  the  odd  num- 
bers. 

429.  Thickness  of  Laminae  for  Newton's  Rings.—  The 
absolute  thickness,  b  e,  d  g,  &c.,  can  also  be  obtained,  a  f  being 
known,  since 

af:ae::ae:ahoTbemy 

for  in  so  short  arcs  the  chord  may  be  considered  equal  to  the  sine, 
that  is,  the  radius  of  the  ring. 

With  air  between  the  lenses  Newton  found  the  thickness  of 
the  first  bright  ring  of  orange-yellow  light,  to  be  1  ^  6*0  0  6  of  an 
inch.  Twice  this,  plus  £  a  wave  length,  must  equal  one  whole 
wave  length,  or 

+     =  Z,  whence 


=  -  000022. 

When  air  is  between  the  lenses,  all  the  rings  range  between 
the  thickness  of  half  a  millionth  of  an  inch  and  72  millionths  ;  if 
water  is  used,  the  limits  are  -f  of  a  millionth  and  58  millionths. 
Below  the  smaller  limit  the  medium  appears  black,  or  no  color  is 
reflected  ;  above  the  highest  limit  the  medium  appears  white,  all 
colors  being  reflected  together.  When  water  is  substituted  for 
air,  all  the  rings  contract  in  diameter,  indicating  that  a  particular 
order  of  color  requires  less  thickness  of  water  than  of  air  ;  the 
thicknesses  for  different  media  are  found  to  be  in  the  inverse 
ratio  of  the  indices  of  refraction. 


284 


OPTICS. 


430.  Relation  of  Rings  by  Reflection  and  by  Trans- 
mission.— If  the  eye  is  placed  beyond  the  lenses,  the  transmitted 
light  also  is  seen  to  be  arranged  in  very  faint  rings,  the  brightest 
portions  being  at  the  same  thicknesses  as  the  darkest  ones  by 
reflection ;  and  these  thicknesses  are  as  the  even  numbers,  2,  4,  6, 
&c.     The  centre,  when  black  by  reflection,  is  white  by  transmis- 
sion, and  where  red  appears  on  one  side,  blue  is  seen  on  the  other  ; 
and,  in  like  manner,  each  color  by  reflection  answers  to  its  comple- 
mentary color  by  transmission,  according  to  Article  427. 

431.  Newton's  Rings  by  a  Monochromatic  Lamp. — 

The  number  of  reflected  rings  seen  in  common  light  is  not  usually 
greater  than  from  five  to  ten.  The  number  is  thus  small,  because 
as  the  outer  rings  grow  narrower  by  a  more  rapid  separation  of 
the  surfaces,  the  different  colors  overlap  each  other,  and  produce 
whiteness.  But  if  a  light  of  only  one  color  falls  on  the  lenses,  the 
number  may  be  multiplied  to  several  hundreds ;  the  rings  are 
alternately  of  that  color  and  black,  growing  more  and  more  narrow 
at  greater  distances,  till  they  can  be  traced  only  by  a  microscope. 
A  good  light  for  such  a  purpose  is  the  flame  of  an  alcohol  lamp, 
whose  wick  has  been  soaked  in  strong  brine,  and  dried. 

432.  Diffraction  or  Inflection. — One  of  the  forms  of  inflec- 
tion is  explained  as  follows  :  Through  an  opaque  screen,  A  B 
(Fig.  272),  let  there  be  a  very  narrow  aperture,  c  d,  by  which  is 

admitted  the  beam  of  red 

10.  &1&.  light,  e  f  g  h,  emanating 

from  a  single  point. 

All  points  of  the  wave 
front  c  d  are  radiants, 
from  which  emanate 
waves  in  all  directions. 
The  original  wave  will 
move  on  through  the 
aperture  forming  the 
band  of  light  g  all.  A 
ray  from  c  as  a  radiant  will  interfere  with  that  ray  from  0,  which 
is  one-half  wave  length  in  advance  of  it,  and  will  give  darkness  at 
some  point  j ;  all  rays  parallel  to  c  j  emanating  from  points 
between  c  and  0,  will  interfere  with  corresponding  rays  from 
points  between  o  and  d,  parallel  to  oj,  and  the  total  result  will 
be  the  dark  bandy  Tc. 

In  like  manner  rays  from  those  points  which  differ  by  f  wave 
lengths  will  interfere  and  produce  the  second  dark  band  m  n. 
Between  these  will  be  a  bright  band,  Jc  m,  due  to  waves  which 
differ  by  one  whole  wave  length.  The  obliquity  of  the  rays  in  the 


INFLECTION    OF    LIGHT.  £85 

figure  has  been  very  greatly  exaggerated,  that  the  lines  may  not 
be  confused.  The  general  plan  of  the  determination  of  the  wave 
length  of  the  color  used  may  be  made  plain  by  reference  to  the 
figure.  Let  o  i  be  drawn  perpendicular  to  c  j.  Then  in  the 
triangle  c  o  i  we  have  c  i  =  c  o  x  cos  o  c  i  —  £  c  d  x  sine/  c  g. 

Now,  because  the  screen  is  at  a  great  distance  from  the  slit  as 
compared  with  c  d,  which  is  about  -^  °f  an  inch,  and  as  g  c  j  is 
very  small  indeed,  about  1.5',  we  have  j  o=j  i,  and  hence 
cj  —  oj  =  cj  —  ij  =  ci',  therefore,  in  order  to  find  c  i  (=  \  a 
wave  length)  we  must  measure  c  d  and  the  angular  deviation  g  cj. 
Instead  of  a  single  aperture,  c  d,  a  great  number  of  very  fine 
parallel  lines  ruled  on  glass  are  used,  and  details  of  measurement 
are  adopted,  a  description  of  which  is  beyond  our  limits.  With 
white  light,  prismatic  fringes  would  be  produced. 

433.  Inflection  by  One  Edge  of  an  Opaque  Body. — If 

one  side  of  the  aperture  g  h  in  the  last  paragraph  be  removed,  the 
effect,  while  due  to  the  same  cause,  will  be  somewhat  modified. 
Let  a  convex  lens  converge  sunlight  to  a  focus  from  which  it 
again  diverges,  the  room  being  dark.  If  we  introduce  into  the 
divergent  pencil  any  opaque  body,  as  a  knife-blade,  for  example,  and 
observe  the  shadow  which  it  casts  on  a  white  screen,  we  shall  ob- 
serve on  both  sides  of  the  shadow  fringes  of  colored  light,  the  differ- 
ent colors  succeeding  each  other  in  the  order  of  the  spectrum,  from 
violet  to  red.  Three  or  four  series  can  usually  be  discerned,  the 
one  nearest  to  the  shadow  being  the  most  complete  and  distinct, 
and  the  remoter  ones  having  fewer  and  fainter  colors.  The  phenom- 
enon is  independent  of  the  density  or  thickness  of  the  body  which 
casts  the  shadow.  The  light,  in  passing  by  the  edge  or  back  of  a 
knife,  by  a  block  of  marble  or  a  bubble  of  air  in  glass,  is  in  each 
case  affected  in  the  same  way.  But  if  the  body  is  very  narrow,  as, 
for  example,  a  fine  wire,  a  modification  arises  from  the  light  which 
passes  the  opposite  side ;  for  now  fringes  appear  within  the 
shadow,  and  at  a  certain  distance  of  the  screen  the  central  line  of 
the  shadow  is  the  most  luminous  part  of  it. 

If  light  of  one  color  be  used,  and  the  distance  of  the  color 
from  the  edge  of  the  shadow  be  measured  when  the  screen  is 
placed  at  different  distances  from  the  body,  it  will  be  found  that 
the  distances  from  the  shadow  are  not  proportional  to  the  distances 
of  the  screen  from  the  body ;  which  proves  that  the  color  is  not 
propagated  in  a  straight  line,  but  in  a  curve.  These  curves  are 
found  to  be  hyperbolas,  having  their  concavity  on  the  side  next 
the  shadow,  and  are  in  fact  a  species  of  caustics. 

434.  Light  through  Small  Apertures. — The  phenomena 
of  inflection  are  exhibited  in  a  more  interesting  manner  when  we 


286  OPTICS. 

view  with  a  magnifying  glass  a  pencil  of  light  after  it  has  passed 
through  a  small  aperture.  For  instance,  in  the  cone  already 
described  as  radiating  from  the  focus  of  a  lens  in  a  dark  room,  let 
a  plate  of  lead  be  interposed,  having  a  pin-hole  pierced  through 
it,  and  let  the  slender  pencil  of  light  which  passes  through  the 
pin-hole  fall  on  the  magnifier.  The  aperture  will  be  seen  as  a 
luminous  circle  surrounded  by  several  rings,  each  consisting  of  a 
prismatic  series.  These  are,  in  truth,  the  fringes  formed  by  the 
edge  of  the  circular  puncture,  but  they  are  modified  by  the  cir- 
cumstance that  the  opposite  edges  are  so  near  to  each  other.  If, 
now,  the  plate  be  removed,  and  another  interposed  having  two 
pin-holes,  within  one-eighth  of  an  inch  of  each  other,  besides  the 
colored  rings  round  each,  there  is  the  additional  phenomenon  of 
long  lines  crossing  the  space  between  the  apertures;  the  lines  are 
nearly  straight,  and  alternately  luminous  and  dark,  and  varying 
in  color,  according  to  their  distance  from  the  central  one.  These 
lines  are  wholly  due  to  the  overlapping  of  two  pencils  of  light, 
for  on  covering  one  of  the  apertures  they  entirely  disappear. 
By  combining  circular  apertures  and  narrow  slits  in  various 
patterns  in  the  screen  of  lead,  very  brilliant  and  beautiful  effects 
are  produced. 

435.  Why  Inflection  is  not  Always  Noticed  in  Look- 
ing by  the  Edges  of  Bodies. — It  must  be  understood  that 
light  is  always  inflected  when  it  passes  by  the  edges  of  bodies ; 
but  that  it  is  rarely  observed,  because,  as  light  comes  from  various 
sources  at  once,  the  colors  of  each  pencil  are  overlapped  and  re- 
duced to  whiteness  by  those  of  all  the  others.  By  using  care  to 
admit  into  the  eye  only  isolated  pencils  of  light,  some  cases  of 
inflection  may  be  observed  which  require  no  apparatus.  If  a  per- 
son standing  at  some  distance  from  a  window  holds  close  to  his  eye 
a  book  or  other  object  having  a  straight  edge,  and  passes  it  along 
so  as  to  come  into  apparent  coincidence  with  the  sash-bars  of  the 
window,  he  will  notice,  when  the  edge  of  the  book  and  the  bar 
are  very  nearly  in  a  range,  that  the  latter  is  bordered  with  colors, 
the  violet  extremity  of  the  spectrum  being  on  the  side  of  the  bar 
nearest  to  the  book,  and  the  red  extremity  on  the  other  side. 
Again,  the  effect  produced  when  light  passes  through  a  narrow  aper- 
ture may  be  seen  by  looking  at  a  distant  lamp  through  the  space 
between  the  bars  of  a  pocket-rule,  or  between  any  two  straight 
edges  brought  almost  into  contact.  On  each  side  of  the  lamp  are 
seen  several  images  of  it,  growing  fainter  with  increased  distance, 
and  finely  colored.  An  experiment  still  more  interesting  is  to 
look  at  a  distant  lamp  through  the  net-work  of  a  bird's  feather. 
There  are  several  series  of  colored  images,  having  a  fixed  arrange- 


PHOSPHORESCENCE.  28? 

ment  in  relation  to  the  disposition  of  the  minute  apertures  in  the 
feather;  for  the  system  of  images  revolves  just  as  the  feather 
itself  is  revolved. 

436.  Length  and  Number  of  Luminous  Waves.— The 
careful  measurements  which  have  been  made  in  cases  of  interfer- 
ence have  led,  by  many  independent  methods,  to  the  accurate 
determination  of  the  length  of  a  wave  of  each  color.  When  the 
length  of  a  wave  of  any  color  is  known,  the  number  of  vibrations 
per  second  is  readily  obtained  by  dividing  the  velocity  of  light  by 
the  length  of  the  wave,  for  light  moves  a  distance  equal  to  one 
wave  length  during  one  vibration  (Art.  421) ;  therefore  if  we  divide 
the  distance  per  second,  186,300  miles,  by  the  length  of  one  wave, 
we  have  the  number  of  vibrations  per  second  as  above. 

The  results  of  these  investigations  give  for  the 


Wave  Length  in 
Inches. 

Number  of  Vibrations  per 
Second. 

Red  

.0000256 

461  000  000,000,000 

Violet  

.0000174 

678,000,000,000,000 

.0000225 

525,000,000,000,000 

437.  Calorescence    and    Fluorescence.  —  Rays  of  less 
refrangibility  than  the  extreme  red  are  due  to  vibrations  too  slow 
to  effect  vision,  but  by  their  great  number  they  possess  great  heat- 
ing power.     If  a  beam  of  light  be  allowed  to  fall  upon  a  thin 
\SLJQJ:  of  a  solution  of  iodine  contained  in  a  suitable  cell,  all  light 
rays  will  be  absorbed,  and  nearly  all  the  rays  of  slow  vibration 
will  pass  through.     These,  if  brought  to  a  focus,  will  communi- 
cate to  refractory  substances   vibrations  sufficiently  rapid  to  be 
recognized  by  the  organ  of  vision,  the  substance  becoming  heated 
to  whiteness. 

Other  rays,  of  vibrations  too  rapid  to  be  recognized  as  light, 
arc  also  found  in  the  spectrum  far  beyond  the  violet.  If  these  are 
allowed  to  fall  upon  a  solution  of  sulphate  of  quinine,  or  upon 
paper  impregnated  with  aesculine,  or  upon  other  substances 
capable  of  reducing  the  rate  of  vibration,  these  substances  become 
visibl?,  glowing  with  a  color  peculiar  to  each  solution,  and  deter- 
mined by  the  rate  of  vibration  which  has  resulted.  This  property 
of  substances  by  which  the  ultra  violet  rays  are  made  visible  is 
called  Fluorescence. 

438.  Phosphorescence. — Very  many  substances,  if   they 
are  exposed  to  a  strong  light  and  then  are  transferred  to  a  dark 
chamber,  continue  to  emit  light  for  longer  or  shorter  periods, 
depending  upon  the  substance  used.    The  sulphides  of  calcium 


288  OPTICS. 

and  strontium  remain  luminous  for  hours  after  exposure  to  sun- 
light. 

Many  other  substances  possess  the  property  of  phosphorescence 
in  so  slight  a  degree,  that  they  will  emit  light  for  only  a  fraction 
of  a  second  after  being  withdrawn  from  the  sun's  rays,  while 
many  seem  not  to  possess  it  at  all. 

What  is  called  phosphorescence  in  certain  animals  and  in 
decaying  animal  and  vegetable  substances  has  no  relation  to  that, 
just  described,  and  can  not  properly  be  considered  here. 


CHAPTER   VII. 

DOUBLE    REFRACTION    AND    POLARIZATION. 

439.  Change  of  Vibrations  in  Polarized  Light. — It  has 
been  stated  (Art.  422)  that  the  vibrations  of  the  ether  in  the  case 
of  common  light,  must  be  supposed  to  be  transverse  in  all  direc- 
tions.    But,  instead  of  this,  we  may  conceive,  what  is  mechani- 
cally equivalent  to  it,  that  the  vibrations  are  made  in  two  trans- 
verse directions  at  right  angles  to  each  other,  and  to  the  direction 
of  the  ray. 

This  being  the  nature  of  common  light,  it  is  easy  to  state  what 
is  meant  by  polarized  light.    It  is  that  in  which  the  vibrations  are 
performed  in  only  one  of  the  transverse  directions.      It  is,  of 
course,  immaterial  what  particular  transverse  motion  is  cut  off, 
provided  all  the  motion  at  right  angles  to  it  is  retained. 

440.  Polarizing  and  Analyzing  by  Reflection. — When 
light  is  reflected,  those  vibrations  of  the  ray  which  are  in  the 
plane  of  incidence  are  generally  weakened  in  a  greater  or  less 
degree,  while  those  which  are  perpendicular  to  the  same  plane  are 
not  affected.     How  much  the  vibrations  are  weakened  depends  on 
the  elasticity  of  the  ether  within  the  medium,  and  on  the  angle 
of  incidence.     But  reflection  of  light  rarely  if  ever  takes  place 
without  diminishing  the  amplitude  of  those  vibrations  which  are 
in  the  plane  of  incidence ;  so  that  a  reflected  ray  is  always  polar- 
ized, at  least,  in  a  slight  degree. 

441.  Polarization  by  Reflection. — Let  two  tubes,  M .2V and 
N  P  (Fig.  273)  be  fitted  together  in  such  a  manner  that  one  can 
be  revolved  upon  the  other;  and  to  the  end  of  each  let  there  be 
attached  a  plate  of  dark-colored  glass,  A  and  C,  capable  of  reflect- 


CHANGES    OP    INTENSITY    DESCRIBED.         289 

ing  only  from  the  first  surface.  These  plates  are  hinged  so  as  to- 
be  adjusted  at  any  angle  with  the  axis  of  the  tube.  Let  the  plane 
of  each  glass  incline  to  the  axis  of  the  tube  at  an  angle  of  33% 
and  let  the  beam  R  A  make  an  incidence  of  57°,  the  complement 

FIG,  273. 


of  33°,  on  A  ;  then  it  will,  after  reflection,  pass  along  the  axis  of 
the  tube,  and  make  the  same  angle  of  incidence  on  C.  If  now 
the  tube  N  P  be  revolved,  the  second  reflected  ray  will  vary  its 
intensity,  according  to  the  angle  between  the  two  planes  of  inci- 
dence on  A  and  C.  The  beam  A  G  is  polarized  light ;  the  glass 
A,  which  has  produced  the  polarization,  is  called  the  polarizing 
plate  ;  the  glass  (7,  which  shows,  by  the  effects  of  its  revolution, 
that  A  C  is  polarized,  is  the  analyzing  plate ;  and  the  whole  in- 
strument, constructed  as  here  represented,  or  in  any  other  manner 
for  the  same  purpose,  is  called  a  polariscope. 

442.  Changes  of  Intensity  Described.— The  changes  in 
the  ray  C  E  are  as  follows  : 

Since  all  vibrations  except  those  perpendicular  to  the  plane  of 
incidence  have  been  destroyed,  when  the  tube  N  P  is  placed  so 
that  the  plane  of  incidence  on  C  is  coincident  with  the  former 
plane  of  incidence,  RAG,  whether  C  E  is  reflected  forward  or 
backward  in  that  plane,  the  intensity  at  E  will  be  the  same  as  if 
A  C  had  been  a  beam  of  common  light.  If  N  P  is  revolved, 
E  will  begin  to  grow  fainter,  and  reach  its  minimum  of  intensity 
when  the  planes  R  A  C  and  A  C  E  are  at  right  angles,  which  is 
the  position  indicated  in  the  figure  ;  for  only  vibrations  at  right 
angles  to  the  plane  of  incidence  can  be  reflected,  and  there  are  no 
such  vibrations  with  reference  to  this  second  plane  of  incidence. 

Continuing  the  revolution,  we  find  the  intensity  increasing 
through  the  second   quadrant  of  revolution,    and   reaching  its 
maximum,  when  the  two  planes  of  incidence  again  coincide,  180° 
from  the  first  position. 
19 


290  OPTICS. 

No  reflection  polarizes  perfectly,  and  hence  there  will  be 
increase  and  decrease  of  the  intensity  of  the  reflected  ray,  without 
total  extinction. 

443.  The  Polarizing  Angle.— The  angle  of  57°  is  called 
the  polarizing  angle  for  glass,  not  because  glass  will  not  polarize 
at  other  angles  of  incidence,  but  because  at  all  other  angles  it 
polarizes  the  light  in  a  less  degree ;  and  this  is  indicated  by  the 
fact  that,  in  revolving  the  analyzing  plate,  there  is  less  change  of 
intensity,  and  the  light  at  E  does  not  become  so  faint.  Different 
substances  have  different  polarizing  angles,  and  for  that  angle  of 
incidence  for  any  substance  which  will  produce  a  maximum  of 
polarization,  the  reflected  and  refracted  rays  will  make  with  each 
other  an  angle  of  90°.  Hence  the  refractive  power  of  opaque 
bodies  may  be  determined.  The  polarization  produced  by  reflec- 
tion from  the  metals  is  very  slight. 

444.  Polarization  by  a  Bundle  of  Plates.  —  Light  may 
also  be  polarized  by  transmission  through  a  bundle  of  laminae  of 
a  transparent  substance,  at  an  angle  of  incidence  equal  to  its 
polarising  angle. 

Since  the  reflected  ray  in  perfectly  polarized  light  consists  of 
vibrations  only  at  right  angles  to  the  plane  of  incidence,  the 
transmitted  light,  being  deprived  of  these  vibrations,  will  consist 
of  vibrations  only  in  the  plane  of  incidence  and  refraction.  As 
no  single  reflection  perfectly  sifts  out  the  vibrations  at  right 
angles  to  the  planes  of  incidence  and  reflection,  many  reflections 
at  successive  surfaces  are  secured,  so  that  finally  there  will  remain 
in  the  refracted  ray  only  vibrations  in  the  plane  of  incidence  and 
refraction. 

Let  a  pile  of  twenty  or  thirty  plates  of  transparent  glass,  no 
matter  how  thin,  be  placed  in  the  same  position  as  the  reflector 
A,  in  Fig.  273,  and  a  beam  of  light  be  transmitted  through  them 
in  a  direction  toward  O.  In  entering  and  leaving  the  bundle  A, 
situated  as  in  the  figure,  the  angles  of  incidence  and  refraction 
are  in  a  horizontal  plane.  When  C  is  revolved,  the  beam  under- 
goes the  same  changes  as  before,  with  this  difference,  that  the 
places  of  greatest  and  least  intensity  will  be  reversed.  If  the  light 
is  reflected  from  G  in  the  same  plane  in  which  it  was  refracted  by 
A,  its  intensity  is  least,  and  it  is  greatest  when  reflected  in  a 
plane  at  right  angles  to  it,  as  at  E  in  the  figure. 

445.  Polarization  by  Absorption. — The  third  and  most 
perfect  method  of  polarizing  light,  is  by  transmission  through 
certain  crystals.  Some  crystals  polarize  the  transmitted  light 
by  absorption.  If  a  thin  plate  be  cut  from  a  crystal  of  tour- 


DOUBLE    REFRACTION.  291 

maline,  by  planes  parallel  to  its  axis,  the  beam  transmitted 
through  it  is  polarized,  the  vibrations  parallel  to  the  axis  being 
transmitted  and  those  perpendicular  to  the  axis  being  absorbed, 
and,  when  received  on  the  analyzing  plate,  will  alternately  be- 
come bright  and  faint,  as  the  tube  of  the  analyzer  is  revolved. 

If  the  analyzer  be  a  plate  of  tourmaline  similar  to  the  polarizer 
the  rays  will  pass  when  the  axes  of  the  plates  are  parallel,  but  will 
fae  wholly  absorbed  when  the  axes  are  at  right  angles  to  each  other. 

446.  Double  Refraction. — There  are  many  transparent  sub- 
stances, particularly  those  of  a  crystalline  structure,  which,  instead 
of  refracting  a  beam  of  light  in  the  ordinary  mode,  divide  it  into 
two  beams.  This  effect  is  called  double  refraction,  and  substances 
which  produce  it  are  called  doubly-refracting  substances. 

This  phenomenon  was  first  observed  in  a  crystal  of  carbonate 
of  lime,  denominated  Iceland  spar.  It  is  bounded  by  six  rhom- 
boidal  faces,  whose  inclinations  to  each  other  are  either  105°  5',  or 
74°  55'.  There  are  two  opposite  solid  angles,  A  and  X  (Fig.  274), 
-each  of  which  is  formed  by  the  meeting  of  three 
obtuse  plane  angles.  A  line  drawn  in  such  direc-  Fl°- 
tion  as  to  be  equally  inclined  to  the  three  edges 
of  either  of  these  obtuse  solid  angles,  is  called  the 
axis  of  the  crystal. 

If  the  edges  of  the  crystal  were  equal,  the  axis 
would  be  the  diagonal  A  X  of  the  rhomb. 

If  a  thick  crystal  of  spar  be  laid  on  a  line  of 
writing,  it  appears  as  two  lines,  one  of  which  seems  not  only 
thrown  aside  from  the  other,  but  brought  a  little  nearer  to  the 
eye;  and  if  the  crystal  be  revolved  in  the  plane  of  the  paper,  and 
the  eye  be  placed  vertically  over  it,  one  image  of  a  letter  will  be 
seen  to  remain  stationary  while  the  other  will  revolve  about  it. 

Therefore  every  ray  of  light,  in  passing  through,  is  divided 

into  two  rays,  which  come  to  the  eye  in  different  directions.    The 

double  refraction  may  also  be  seen  by  letting  a 

'IG.  275.  yery  glender  sunbeam>  R  r  (pig.  275),  fall  on 

the  crystal ;  as  it  enters  it  takes  two  directions, 
r  0,  and  r  E,  which  on  passing  out  describe 
the  lines  0  0',  E  E',  parallel  to  the  incident 
beam,  R  r.  One  of  these  rays,  0  0',  is  called 
the  ordinary  ray,  because  it  is  always  refracted 
according  to  the  ordinary  law.  of  refraction 
(Art.  369) ;  that  is,  it  remains  in  the  plane  of 
incidence,  and  the  sines  of  incidence  and  re- 
fraction have  a  constant  ratio  to  each  other  at 
all  inclinations.  The  other,  E  E',  is  called  the 


292  OPTICS. 

extraordinary  ray,  because  in  some  positions  it  departs  from  this 
law  of  refraction  in  one  or  both  particulars. 

In  the  experiment  above,  the  ordinary  image  seems  nearest  to 
the  eye,  and  is  stationary,  when  viewed  vertically,  while  the 
crystal  revolves. 

The  property  of  double  refraction  belongs  to  a  large  number 
of  crystals,  and  also  to  some  animal  substances,  as  hair,  quills, 
&c.;  and  it  may  be  produced  artificially  in  glass  by  heat  or 
pressure. 

447.  Optical  Relations  of  the  Axis. — The  axis  of  Iceland 
spar  has  been  defined  with  reference  to  form  ;  but  it  is  also  the 
axis  with  respect  to  its  optical  relations,  for  in  the  direction  of 
that  line  a  ray  is  never  doubly  refracted,  while  it  is  doubly  re- 
fracted in  all  other  directions. 

Every  plane  which  includes  the  axis  of  a  crystal  is  called  a 
principal  section.  In  every  principal  section  the  extraordinary 
ray  conforms  to  one  part  of  the  law  of  refraction,  but  not  to  the 
other ;  it  remains  in  the  plane  of  incidence,  but  does  not  preserve 
a  constant  ratio  of  sines  at  different  inclinations. 

In  a  plane  at  right  angles  to  the  axis,  the  extraordinary  ray 
conforms  to  both  parts  of  the  law  ;  but  in  all  planes  besides  this 
and  the  principal  sections,  it  conforms  to  neither  part. 

Crystals  of  a  positive  axis,  are  those  in  which  the  extraordinary 
ray  has  a  larger  index  of  refraction  than  the  ordinary  ray  ;  crys- 
tals of  a  negative  axis  are  those  in  which  the  index  of  the  extraor- 
dinary ray  is  less  than  that  of  the  ordinary  ray.  Iceland  spar  is  a 
crystal  of  negative  axis. 

Some  crystals  have  two  axes  of  double  refraction  ;  that  is, 
there  are  two  directions  in  which  light  may  be  transmitted  with- 
out being  doubly  refracted.  A  few  crystals  have  more  than  two 


448.  Polarizing  by  Double  Refraction. — In  doubly-re- 
fracting crystals,  the  ether  possesses  different  degrees  of  elasticity 
in  different  directions  ;  hence,  so  far  as  vibrations  lie  in  one  plane, 
they  may  be  more  retarded  in  their  progress,  and  in  a  plane  at 
right  angles  to  that  they  may  be  less  retarded,  and  the  degree 
of  refraction  depends  on  the  amount  of  retardation  (Art.  424). 
Thus  the  two  systems  become  separated,  and  emerge  at  different 
places. 

If  a  beam  is  passed  through  a  doubly-refracting  crystal,  and 
the  two  parts  fall  on  the  analyzing  plate,  they  will  come  to  their 
points  of  greatest  and  least  brightness  at  alternate  quadrants ; 
indeed,  when  one  ray  is  brightest,  the  other  is  entirely  extin- 


DIFFERENT    KINDS    OF    POLARIZATION.       293 

guished.  Therefore  the  two  rays  which  emerge  from  a  doubly- 
refracting  crystal  are  polarized  completely,  the  ordinary  ray  in  a 
principal  plane  and  the  extraordinary  ray  in  a  plane  at  right 
angles  to  a  principal  plane. 

449.  Different  Kinds  of   Polarization. — Since  the  dis- 
covery was  made  that  the  ethereal  atoms  may  by  certain  methods 
be  thrown  into  circular  movements,  and  by  others  into  vibrations 
in  an  ellipse  with  the  axis  in  a  fixed  direction,  the  polarization 
already  described   has  been  called  plane  polarization,  since  the 
atoms  vibrate  in  a  plane.     Circular  polarization  is  that  in  which 
the  atoms  revolve  in  circles  ;  and  elliptical  polarization  denotes  a 
state  of  vibration  in  ellipses,  whose  major  axes  are  confined  to 
one  plane. 

The  consideration  of  these  various  modes  of  polarization  de- 
mands more  space  than  can  be  spared  here. 

450.  Every  Polarizer  an  Analyzer. — We  have  seen  that 
light  is  polarized  by  reflection  from  glass  at  an  incidence  of  57°, 
and  analyzed  by  another  plate  at  the  same  angle  of  incidence. 
This  is  but  an  instance  of  what  is  always  true,  that  every  method 
of  polarizing  light  may  be  used  to  analyze,  i.  e.,  to  test  its  polar- 
ization. '  Hence,  a  bundle  of  thin  plates  of  glass  may  take  the 
place  of  the  analyzer  C,  as  well  as  of  the  polarizer  A.     For,  on 
turning  it  round,  though  the  transmitted  beam  remains  in  the 
same  place,  yet  it  will,  at  the  alternate  quadrants,  brighten  to  its 
maximum  and  fade  to  its  minimum  of  intensity. 

So,  again,  if  light  has  passed  through  a  tourmaline,  and  is 
received  on  a  second  whose  crystalline  axis  is  parallel  to  that  of 
the  former,  the  ray  will  proceed  through  that  also;  but  if  the 
second  is  turned  in  its  own  plane,  the  transmitted  ray  grows  faint, 
and  nearly  disappears  at  the  moment  when  the  two  axes  are  at  90° 
of  inclination,  and  this  alternation  continues  at  each  90°  of  the 
whole  revolution. 

Finally,  place  a  double-refractor  at  each  end  of  the  polar i  scope, 
and  let  a  beam  pass  through  them  and  fall  on  a  screen.  The  first 
crystal  will  polarize  each  ray,  and  the  second  will  doubly  refract 
and  also  analyze  each,  exhibiting  a  very  interesting  series  of 
changes.  In  general,  four  rays  will  emerge  from  the  second 
crystal,  producing  four  luminous  spots  on  the  screen.  But,  on 
revolving  the  tube,  not  only  do  the  rays  commence  a  revolution 
round  each  other,  but  two  of  them  increase  in  brightness,  and  the 
other  two  at  the  same  time  diminish  as  fast,  till  two  alone  are 
visible,  at  their  greatest  intensity.  At  the  end  of  the  second 
quadrant,  the  spots  before  invisible  are  at  their  maximum  of 
brightness,  and  the  others  are  extinguished.  '  This  alternation 


294  OPTICS. 

continues  as  long  as  the  crystal  is  revolved.    In  the  middle  of 
each  quadrant  the  four  are  of  equal  brightness. 

451.  Nicol's  Prism. — As  the  four  beams  in  the  last  case 
are  an  annoyance  in  investigations  requiring  polarized  light,  only 
one  is  retained,  the  others  being  turned  aside. 

A  rhomb  of  Iceland  spar  is  cut  by  a  plane  A  B  X  passing 

through  the  obtuse  angles  (Fig.  274) ;  the  two  halves  are  then 

FIG  276  joined  together  again, 

as  they  were  before  the 
cutting,  by  Canada 
balsam.  This  is  a 
Nicol  prism.  If  a 
beam  enters  the  prism 
at  a  (Fig.  276),  it  will 
be  separated  into  two 
beams,  the  ordinary  and  the  extraordinary,  and  then  will  fall 
upon  the  film  of  Canada  balsam  C  B.  Now  the  refractive  index 
of  Canada  balsam  is  less  than  the  index  of  refraction  for  the 
ordinary  ray,  and  greater  than  that  for  the  extraordinary  ray ; 
hence  the  ordinary  ray  will  be  totally  reflected  at  o,  and  will  pass 
out  at  the  side  of  the  prism,  while  the  extraordinary  ray  will  be 
refracted  through  the  film  of  balsam,  emerging  as  polarized  light 
at  m.  If  a  similar  prism  be  used  as  an  analyzer,  one  of  the  two 
rays  into  which  the  polarized  ray  is  separated  is  again  turned  out 
of  the  prism  by  total  reflection,  and  only  a  single  ray  emerges. 

452.  Color  by  Polarized  Light. — The  phenomena  of  color 
produced  by  polarized  light  are  beautiful,  and  of  great  interest. 

Let  a  thin  principal  section  of  some  doubly-refracting  crystal 
be  placed  perpendicularly  across  the  axis  of  the  polariscope  (Fig. 
277),  whose  analyzer  is  turned 
so  that  the  reflected  ray  C  E 
is  at  its  minimum  intensity. 
When  the  axis  of  the  crystal, 
D  H,  coincides  with  the  first 
plane  of  reflection,  R  A  C,  or 
is  perpendicular  to  it,  all  the 
phenomena  are  the  same  as  if 
no  crystal  was  interposed.  But  let  the  film  be  revolved  in  its 
own  plane  till  D  H  makes  45°  with  the  plane  R  A  C ;  then,  in- 
stead of  the  dark  spot  at  E,  a  brilliant  color  appears.  That  color 
may  be  any  tint  of  the  spectrum,  according  to  the  thickness  of 
the  interposed  film.  If  now  the  revolution  of  the  crystal  is  con- 
tinued, the  color  fades  out  at  the  end  of  the  next  45°,  reappears 


SYSTEMS    OF    COLORED    RINGS.  395 

at  90°,  and  so  on.  But  if  the  crystal  be  so  placed  as  to  give 
color,  and  the  analyzing  plate  be  revolved,  a  different  series  pre- 
sents itself.  The  color  observed  at  E,  during  the  first  45°,  grad- 
ually fades,  and  during  the  next  45°  its  complement  appears  and 
brightens  to  its  maximum.  The  original  color  is  restored  at  180°, 
and  the  complementary  color  at  2709. 

The  most  interesting  form  of  this  experiment  is  seen  when  the 
light  is  polarized  and  analyzed  by  means  of  double-refractors ; 
since  the  polarization  is  more  perfect,  and  the  two  pairs  of  oppo- 
sitely polarized  rays  are  on  the  screen  at  once.  When  two  of  the 
images  are  of  a  certain  color,  the  other  two  have  the  complemen- 
tary color. 

453.  Systems  of  Colored  Rings. — Systems  of  irised  bands 
and  rings  may  also  be  produced  by  the  polariscope.  Let  a  plate 
be  cut  from  a  doubly-refracting  crystal  of  one  axis  by  planes  per- 
pendicular to  that  axis;  and  place  it  between  the  polarizer  and 
analyzer.  If  now  a  pencft  of  sufficient  divergency  is  transmitted, 
a  system  of  colored  circles  will  be  formed,  resembling  Newton's 
rings  between  lenses.  If  a  polariscope  is  formed  of  two  tourma- 
lines, and  the  crystal  laid  between  them,  and  the  whole  combina- 
tion, less  than  half  an  inch  thick,  is  brought  close  to  the  eye,  the 
pencil  of  light  will  consist  of  rays  of  various  obliquity,  and  the 
rings  may  be  seen  beautifully  projected  on  the  sky.  Or  the  ring 
systems  may  be  projected  on  a  screen  by  a  polariscope  furnished 
with  concentrating  lenses.  Fig.  278  presents  the  system  as  seen 
through  Iceland  spar  when  the  planes  of  reflection  in  the  polari- 

FIG.  278.  FIG.  270. 


scope  are  at  right  angles.  Two  dark  diameters  cross  the  system 
and  interrupt  the  rings.  If  the  planes  of  reflection  are  coincident, 
the  system  is  in  every  respect  complementary  to  the  other  (Fig. 
279).  The  colors  of  the  rings  are  all  reversed,  and  the  crossing 
bands  are  white.  If  double-refractors  of  two  axes  are  used  instead 
of  the  spar,  compound  systems  are  shown,  of  various  forms  and 
great  beauty. 

In  treating  of  this  subject  it  has  been  assumed  that  the  vibra- 


296  OPTICS. 

tions  of  the  reflected  polarized  beam  are  at  right  angles  to  the 
plane  of  reflection,  called  also  the  plane  of  polarization  ;  but  this 
assumption  is  made  merely  to  help  the  student  to  fix  the  facts  in 
mind,  since  it  is  a  point  in  regard  to  which  the  theory  is  not  yet 
authoritatively  settled. 


CHAPTER    VIII. 

VISION. 

454.  Image  by  Light  through  an  Aperture.— If  light 
from  an  external  object  pass  through  a  small  opening  of  any  shape 
in  the  wall  of  a  dark  room,  it  will  form  an  ill-defined  inverted 
image   on   the  opposite  wall.     Imagine  a  minute  square  orifice, 
through  which  the  light  enters  and  falls  on  a  screen  several  feet 
distant.     A  pencil  of  light,  in  the  shape  of  a  square  pyramid, 
emanating  from  the  highest  point  of  the  object,  passes  through 
the  aperture,  and  forms  a  luminous  square  near  the  bottom  of  the 
screen.     From  an  adjacent  point  another  pencil,  crossing  the  first 
at  the  aperture,   forms   another  square,  overlapping  and  nearly 
coinciding  with  the  former.     Thus  every  point  of  the  object  is 
represented  by  its  square  on  the  screen ;  and  as  the  pencils  all 
cross  at  the  aperture,  the  image  formed  is  every  way  inverted.    It 
is  also  indistinct,  because  the  squares  overlap,  and  the  light  of 
contiguous  points  is  mingled  together.     If  the  orifice  is  smaller, 
the  image  is  less  luminous,  but  more  distinct,  because  the  pencils 
which  form  it  overlap  in  a  less  degree.     If  the  hole  is  circular,  or 
triangular,  or  of  irregular  form,  there  is  no  change  in  the  appear- 
ance of  the  image,  which  is  now  composed  of  small  circles,  or 
triangles,  or  irregular  figures,  whose  shape  is  completely  lost  by 
overlapping. 

455.  Effect  of  a  Convex  Lens  at  the  Aperture. — The 

image  will  become  distinct,  and  more  luminous  also,  if  the  aper- 
ture be  enlarged  to  a  diameter  of  two  or  three  inches,  and  then 
covered  by  a  convex  lens  of  the  proper  curvature.  The  image  will 
be  distinct,  because  the  rays  from  each  point  of  the  object  are  con- 
verged to  a  point  again,  and  luminous  in  proportion  as  the  lens 
has  a  larger  area  than  the  aperture  before  employed.  This  is  a 
real,  and  therefore  an  inverted  image  (Art.  385).  A  scioptic  ball 
is  a  sphere  containing  a  lens,  and  so  fitted  in  a  socket  that  it  can 
be  turned  in  any  direction,  and  thus  bring  into  the  room  the 


THE    EYE. 


297 


PIG.  280 


images  of  different  parts  of  the  landscape.  The  camera  obscura,  is  a 
darkened  room  furnished  with  a  scioptic  ball  and  adjustable  screen 
for  producing  distinct  pictures  of  external  objects. 

Instead  of  connecting  the  lens  with  the  wall  of  a  room,  it  is 
frequently  attached  to  a  portable  box  or  case,  within  which  the 
image  is  formed.  The  Daguerreotype,  or  photograph,  is  the  image 
produced  by  the  convex  lens,  and  rendered  permanent  by  the 
chemical  action  of  light  on  a  surface  properly  prepared.  The  lens 
for  photographic  purposes  needs  to  be  achromatic,  and  corrected, 
also,  as  far  as  possible,  for  spherical  aberration. 

456.  The  Eye. — The  eye  is  a  camera  obscura  in  miniature  ; 
we  find  here  the  darkened  room,  the  aperture,  the  convex  lens, 
and  the  screen,  with  inverted  images  of  external  objects  painted 
on  it.     A  horizontal  section  of  the  eye  is  represented  in  Fig.  280. 

The  optical  apparatus  of  the  eye,  and  the  spherical  case  which 
incloses  it,  constitute 
what  is  called  the  eye- 
ball The  case  itself, 
except  about  a  sixth 
part  of  it  in  front,  is  a 
strong  white  substance, 
called,  on  account  of  its 
hardness,  the  sclerotic 
coat,  S,  S  (Fig.  280). 
In  the  front,  this  opaque 
coat  changes  to  a  per- 
fectly transparent  cover- 
ing, called  the  cornea, 
C,  C,  which  is  a  little 
more  convex  than  the 
sclerotic  coat.  The  in- 
creased convexity  of  the 

cornea  may  be  felt  by  laying  the  finger  gently  on  the  eyelid  when 
closed,  and  then  rolling  the  eye  one  way  and  the  other. 

The  bony  socket,  which  contains  the  eye,  is  of  pyramidal  form, 
its  vertex  being  some  distance  behind  the  eyeball ;  room  is  thus 
afforded  for  the  mechanism  which  gives  it  motion.  This  cavity, 
except  the  hemisphere  in  front  occupied  by  the  eye  itself,  is  filled 
up  with  fatty  matter  and  with  the  six  muscles  by  which  the  eye- 
ball is  revolved  in  all  directions. 

457.  The  Interior  of  the  Eye. — Behind  the  cornea  is  a 
fluid,  A,  called  the  aqueous  humor.     In  the  back  part  of  this  fluid 
lies  the  iris,  /,  /,  an  opaque  membrane,  having  in  the  centre  of  it 
a  circular  aperture,  the  pupil,  through  which  the   light  enters. 


OF  THE 

UNIVERSITY 


298  OPTICS. 

The  iris  is  the  colored  part  of  the  eye  ;  the  back  side  of  it  is  black. 
Directly  back  of  the  aqueous  humor  and  iris,  is  a  flexible  double 
convex  lens,  L,  called  the  crystalline  lens,  or  crystalline  humor, 
having  the  greatest  convexity  on  the  back  side.  The  large  space 
back  of  the  crystalline  is  occupied  by  the  vitreous  humor,  F,  a 
semi-liquid,  of  jelly-like  consistency.  Next  to  the  vitreous  humor 
succeed  those  inner  coatings  of  the  eye,  which  are  most  imme- 
diately concerned  in  vision.  First  in  order  is  the  retina,  R,  R,  on 
which  the  light  paints  the  inverted  pictures  of  external  objects. 
The  fibres  of  the  optic  nerve,  which  enter  the  ball  at  N,  are  spread 
all  over  the  retina,  and  convey  the  impressions  produced  there  to 
the  brain.  Outside  of  the  retina  is  the  choroid  coat,  c  h,  c  h, 
covered  with  a  black  pigment,  which  serves  to  absorb  all  the  light 
so  soon  as  it  has  passed  through  the  retina  and  left  its  impressions. 
The  choroid  is  inclosed  by  the  sclerotic  already  described.  The 
nerve-fibres,  which  are  spread  over  the  interior  of  the  retina,  are 
gathered  into  a  compact  bundle  about  one-tenth  of  an  inch  in 
diameter,  which  passes  out  through  the  three  coatings  at  the  back 
part  of  the  ball,  about  fifteen  degrees  from  the  axis,  XX,  on  the 
side  toward  the  other  eye.  M,  M  represent  two  of  the  muscles, 
where  they  are  attached  to  the  eyeball. 

458.  Vision, — The  index  of  refraction  for  the  cornea,  and  the 
aqueous  and  vitreous  humors,  is  just  about  the  same  as  that  for 
water;  for  the  crystalline  lens,  the  index  is  a  little  greater.     The 
light,  therefore,  which  comes  from  without,  is  converged  principally 
on  entering  the  cornea,  and  this  convergency  is  a  little  increased 
both  on  entering  and  leaving  the  crystalline.     If  the  convergency 
is  just  sufficient  to  bring  the  rays  of  each  pencil  to  a  focus  on  the 
retina,  then  the  images  are  perfectly  formed,  and  there  is  distinct 
vision.     To  prevent  the  reflection  of  rays  back  and  forth  within 
the  chamber  of  the  eye,  its  walls  are  made  perfectly  black  through- 
out by  a  pigment  which  lines  the  choroid,  the  ciliary  processes,  and 
the  back  of  the  iris.     Telescopes  and  other  optical  instruments 
are  painted  black  in  the  interior  for  a  similar  purpose. 

The  cornea  is  prevented  from  producing  spherical  aberration 
by  the  form  of  a  prolate  spheroid  which  is  given  to  its  surface, 
and  the  crystalline,  by  a  gradual  increase  of  density  from  its  edge 
to  its  centre. 

459.  Adaptations. — By  the  prominence  of  the  cornea  rays 
of  considerable  obliquity  are  converged  into  the  pupil,  so  that  the 
eye,  without  being  turned,  has  a  range  of  vision  more  or  less  per- 
fect, through  an  angle  of  about  150°. 

The  quantity  of  light  admitted  into  the  eye  is  regulated  by  the 
size  of  the  pupil.  The  iris,  composed  of  a  system  of  circular  ;m'l 


ADAPTATIONS.  299 

radial  muscles,  expands  or  contracts  the  pupil  according  to  the 
intensity  of  the  light.  These  changes  are  involuntary  ;  a  person 
may  see  them  in  his  own  eyes  by  shading  them,  and  again  letting 
a  strong  light  fall  upon  them,  while  he  is  before  a  mirror. 

The  pupils  in  the  eyes  of  animals  have  different  forms  accord- 
ing to  their  habits  ;  in  the  eyes  of  those  which  graze,  the  pupil  is 
elongated  horizontally,  and  in  the  eyes  of  beasts  and  birds  of  prey, 
it  is  elongated  vertically. 

The  eyes  of  animals  are  adapted,  in  respect  to  their  refractive 
power,  to  the  medium  which  surrounds  them.  Animals  which 
inhabit  the  water  have  eyes  which  refract  much  more  than  those 
of  land  animals.  The  human  eye  being  fitted  for  seeing  in  air,  is 
unfit  for  distinct  vision  in  water,  since  its  refractive  power  is 
nearly  the  same  as  that  of  water,  and  therefore  a  pencil  of  parallel 
rays  from  water  entering  the  eye  would  scarcely  be  converged  at 
all.  The  effect  is  the  same  as  if  the  cornea  were  deprived  of  all 
its  convexity. 

460.  Accommodation  to  Diminished  Distance.— It  has 

been  shown  (Art.  385),  that  as  an  object  approaches  a  lens,  its 
image  moves  away,  and  the  reverse.  Therefore  in  the  eye  there 
must  be  some  change  in  order  to  prevent  this,  and  keep  the 
image  distinct  on  the  retina  while  the  object  varies  its  distance. 
In  a  state  of  rest,  the  eye  converges  to  the  retina  only  the  pencils 
of  parallel  rays,  that  is,  those  which  come  from  objects  at  great 
distances.  Rays  from  near  objects  diverge  so  much  that,  while  the 
eye  is  at  rest,  it  cannot  sufficiently  converge  them  so  that  they 
will  meet  on  the  retina ;  but  each  conical  pencil  is  cut  off  before 
reaching  its  focus,  and  all  the  points  of  the  object  are  represented 
by  overlapping  circles,  causing  an  indistinct  image.  The  change 
in  the  eye,  which  fits  it  for  seeing  near  objects  distinctly,  is  called 
accommodation.  This  is  effected  by  increasing  the  convexity  of 
the  crystalline  lens,  principally  the  front  surface.  The  ciliary 
muscle,  m,*m,  surrounds  the  crystalline,  and  is  attached  to  the 
sclerotic  coat  just  on  the  circle  where  it  changes  into  the  cornea. 
This  muscle  is  connected  with  the  edge  of  the  crystalline  by  the 
circular  ligament  which  surrounds  the  latter  and  holds  it  in  place. 
When  the  muscle  contracts,  it  relaxes  the  ligament,  and  the  crys- 
talline, by  its  own  elastic  force,  begins  to  assume  a  more  convex 
form,  as  represented  by  the  dotted  line.  The  eye  is  then  accom- 
modated for  the  vision  of  objects  more  or  less  near,  according  to 
the  degree  of  change  in  the  lens.  On  the  other  hand,  when  the 
ciliary  muscle  relaxes,  the  ligament  again  draws  upon  the  lens  to 
flatten  it,  and  adapt  it  for  the  view  of  distant  objects.  In  Fig. 
281  these  two  conditions  of  the  crystalline  are  more  distinctly 


300 


OPTICS. 


FIG.  281. 


shown.  The  dotted  line  exhibits  the  shape  of  the  lens  when 
accommodated  for  seeing  near  objects.  Accompany- 
ing this  action  of  the  ciliary  muscle  is  that  of  the 
iris,  which  diminishes  the  pupil  for  near  objects,  so 
as  to  exclude  the  outer  and  more  divergent  rays. 
The  dotted  lines  in  front  of  the  iris  represent  its 
situation  when  pushed  forward  by  the  crystalline 
accommodated  for  near  objects. 

461.  Long-Sightedness. — As  life  advances, 
the  crystalline  becomes  harder  and  less  elastic.  It 
therefore  assumes  a  less  convex  form  when  the  liga- 
ment is  relaxed,  and  cannot  be  accommodated  to  so  short  dis- 
tances as  in  earlier  years  ;  and  at  length  it  remains  so  flattened  in 
shape  that  only  very  distant  objects  can  be  seen  distinctly.  The 
eye  is  then  said  to  be  long-sighted,  and  requires  a  convex  lens  to 
be  placed  before  it,  to  compensate  for  insufficient  convexity  in  the 
crystalline. 

There  are,  however,  cases  of  long-sightedness  in  early  life. 
Such  instances  are  found  to  be  the  result  of  an  oblate  form  of  the 
eyeball,  as  shown  in  Fig. 
282;  it  is  too  short  from 
front  to  back  to  furnish 
room  for  the  convergency  of 
the  pencils,  and  they  are 
cut  off  by  the  retina  before 
reaching  their  focal  points. 
In  order  to  bring  the  dis- 
tinct image  forward  upon  the  retina,  convex  glasses  are  needed  in 
such  cases,  just  as  for  the  eyes  of  most  people  when  advanced  in 
life.  As  the  term  long-sightedness  is  now  applied  to  this  abnor- 
mal condition  of  the  eye,  the  effect  of  age  upon  the  sight  is  more 
properly  called  old-sightedness. 


FIG.  282. 


FIG.  283. 


462.  Short-Sightedness. — The  eyes  of  the  short-sighted 

have  a  form  the  reverse  of 
that  just  described;  the  eye- 
ball is  elongated  from  cornea 
to  retina  (Fig.  283),  resem- 
bling a  prolate  spheroid,  so 
that  rays  parallel,  or  nearly 
so,  are  converged  to  a  point 

_^^  before  reaching    the  retina, 

and  after  crossing,  fall  on  it  in  a  circle  ;  and  the  image,  made  up 
of  overlapping  circles  instead  of  points,  is  indistinct.      If  this 


THE    BLIND    POINT.  301 

elongation  of  the  eyeball  is  extreme,  an  object  must  be  brought 
very  near,  in  order  that  its  image  may  move  back  to  the  retina, 
and  distinct  vision  be  produced.  This  inconvenience  is  remedied 
by  the  use  of  concave  lenses,  which  increase  the  divergency  of  the 
rays  before  they  enter  the  eye,  and  thus  throw  their  focal  points 
further  back. 

In  the  normal  condition  of  the  eyes  in  early  life,  the  nearest 
limit  of  distinct  vision  is  about  five  inches.  This  limit  slowly 
increases  with  advance  of  life,  but  much  more  slowly  in  some 
cases  than  others,  till  it  is  at  an  indefinitely  great  distance.  The 
near  limit  of  distinct  vision  for  the  short-sighted  varies  from  Jive 
down  to  two  inches,  according  to  the  degree  of  elongation  in  the 
eyeball. 

463.  Why  an  Object  is  Seen  Erect  and  Single.— The 
image  on  the  retina  is  inverted;  and  that  is  the  very  reason  why 
the  object  is  seen  erect ;  the  image  is  not  the  thing  seen,  but  that 
by  means  of  which  ive  see.     The  impression  produced  at  any  point 
on  the  retina  is  referred  outward  in  a  straight  line  through  a  point 
near  the  centre  of  the  lens,  to  something  external  as  its  cause ; 
and  therefore  that  is  judged  to  be  highest  without  us  which  makes 
its  image  lowest  on  the  retina,  and  the  reverse. 

An  object  appears  as  one,  though  we  see  it  by  means  of  two 
images ;  but  this  is  only  one  of  many  instances  in  which  we  have 
learned  by  experience  to  refer  two  or  more  sensations  to  one  thing 
as  the  cause.  Provided  the  images  fall  on  parts  of  the  retina, 
which  in  our  ordinary  vision  correspond  with  each  other,  then  by 
experience  we  refer  both  impressions  to  one  object ;  but  if  we 
press  one  eye  aside,  the  image  falls  in  a  new  place  in  relation  to 
the  other,  and  the  object  seems  double. 

464.  Indirect  Vision.— The  Blind  Point— To  obtain  a 
clear  and  satisfactory  view  of  an  object,  the  axes  of  both  eyes  are 
turned  directly  upon  it,  in  which  case  each  image  is  at  the  centre 
of  the  retina.     But  when  the  light  from  an  object  is  exceedingly 
faint,  it  is  better  seen  by  indirect  vision,  that  is,  by  looking  to  a 
point  a  little  on  one  side,  and  especially  by  changing  the  direction 
of  the  eyes  from  moment  to  moment,  so  that  the  image  may  fall 
in  various  places  near  the  centre  of  the  retina.     Many  heavenly 
bodies  are  plainly  discerned  by  indirect  vision,  which  are  too  faint 
to  be  seen  by  direct  vision. 

In  the  description  of  the  eye  it  was  stated  that  the  retina,  as 
well  as  the  choroid  and  the  sclerotic,  is  perforated  to  allow  the 
optic  nerve  to  pass  through.  At  that  place  there  is  no  vision,  and 
it  is  called  the  blind  point.  In  each  eye  it  is  situated  about  15° 
from  the  centre  of  the  retina  toward  the  other  eye.  Let  a  person 


302  OPTICS. 

close  his  right  eye,  and  with  the  left  look  at  a  small  but  conspicu- 
ous object,  and  then  slowly  turn  the  eye  away  from  it  toward  the 
right ;  presently  the  object  will  entirely  disappear,  and  as  he  looks 
still  further  to  the  right,  it  will  after  a  moment  reappear,  and  con- 
tinue in  sight  till  the  axis  of  the  eye  is  turned  70°  or  80°  from  it. 
The  same  experiment  may  be  tried  with  the  right  eye  in  the  oppo- 
site direction.  The  reason  why  people  do  not  generally  notice  the 
fact  till  it  is  pointed  out,  is  that  an  object  cannot  disappear  to  both 
eyes  at  once,  nor  to  either  eye  alone,  when  directed  to  the  object. 

465.  Continuance  of  Impressions. — The  impression  which 
a  visible  object  makes  upon  the  retina  continues  about  one-eighth 
or  one-ninth  of  a  second  ;  so  that  if  the  object  is  removed  for 
that  length  of  time,  and  then  occupies  its  place  again,  the  vision 
is  uninterrupted.     A  coal  of  fire  whirled  round  a  centre  at  the 
rate  of  eight  or  nine  times  per  second,  appears  in  all  parts  of  the 
circumference  at  once.     When  riding  in  the  cars,  one  sometimes 
gets  a  faint  but  apparently  an  uninterrupted  view  of  the  land- 
scape beyond  a  board  fence,  by  means  of  successive  glimpses  seen 
through  the  cracks  between  the  upright  boards.     Two  pictures, 
on  opposite  sides  of  a  disk,  are  brought  into  view  together,  as 
parts  of  one  and  the  same  picture,  by  whirling  the  disk  rapidly 
on  one  of  its  diameters.     Such  an  instrument  is  called  a  thauma- 
trope.    The  phantasmascope  is  constructed  on  the  same  principle. 
Several  pictures  are  painted  in  the  sectors  of  a  circular  disk, 
representing  the  same  object  in  a  series  of  positions.     These  are 
viewed  in  a  mirror  through  holes  in  the  disk,  as  it  revolves  quickly 
in  its  own  plane.     Each  glimpse  which  is  caught  whenever  a  hole 
comes  before  the  eye,  presents  the  object  in  a  new  attitude  ;  and 
all  these  views  are  in  such  rapid  succession  that  they  appear  like 
one  object  going  through  the  series  of  movements. 

466.  Subjective   Colors. — There  are  impressions  on  the 
retina  of  another  kind,  which  are  produced  by  intense  lights  ;  they 
continue  longer,  and  are  in  respect  to  color  unlike  the  objects 
which  cause  them.     They  are  called  subjective  colors:    If  a  par- 
ticular part  of  the  retina  is  for  some  time  affected  by  the  image  of 
a  bright  colored  object,  and  then  the  eyes  are  shut,  or  turned 
upon  a  white  surface,  the  form  appears  to  remain,  but  the  color  is 
complementary  to  that  of  the  object;  and  its  continuance  is  for  a 
few  seconds  or  several  minutes,  according  to  the  vividness  of  the 
impression. 

That  portion  of  the  retina  upon  which  the  bright  colored 
image  was  formed  loses  sensitiveness  to  that  particular  color  after 
a  short  time,  and  when  white  light  falls  upon  the  retina  that  par- 
ticular spot  is  affected  by  the  complementary  color  only. 


DISTANCE    OF    BODIES.  303 

467.  Irradiation. — When  small  bodies  are  intensely  illumi- 
nated the  retina  is  affected  somewhat  beyond  their  proper  images 
upon  it,  and  the  bodies  consequently  appear  larger  than  they 
would  if  less  bright.     A  white  circle  upon  a  black  ground  looks 
larger  than  an  equal  black  circle  upon  a  white  ground.     At  new 
moon,  when  both  the  bright  and  the  dark  portions  are  visible,  the 
crescent  seems  to  be  a  part  of  a  larger  sphere  than  that  which  it 
accompanies. 

This  enlargement  of  the  image  is  called  irradiation. 

468.  Estimate  of  the  Distance  of  Bodies.— 

1.  If  objects  are  near,  we  judge  of  relative  distance  by  the  in- 
clination of  the  optic  axes  to  each  other.     The  greater  that  incli- 
nation is,  or,  which  is  the  same  thing,  the  greater  the  change  of 
direction  in  an  object,  as  it  is  viewed  by  one  eye  and  then  by  the 
other,  the  nearer  it  is.     If  objects  are  very  near,  we  can  with  one 
eye  alone  judge  of  their  distance  by  the  degree  of  effort  required 
to  accommodate  the  eye  to  that  distance. 

2.  If  objects  are  known,  we  estimate  their  distance  by  the 
visual  angle  which  they  fill,  having  by  experience  learned  to 
associate  together  their  distance  and  their  apparent,  that  is,  their 
angular  size. 

3.  Our  judgment  of  distant  objects  is  influenced  by  their  clear- 
ness  or  obscurity.    Mountains,  and  other  features  of  a  landscape, 
if  seen  for  the  first  time  when  the  air  is  remarkably  pure,  are 
estimated  by  us  nearer  than  they  really  are;  and  the  reverse,  if 
the  air  is  unusually  hazy. 

4.  Our  estimate  of  distance  is  more  correct  when  many  objects 
intervene.     Hence  it  is  that  we  are  able  to  place  that  part  of  the 
sky  which  is  near  the  horizon  further  from  us  than  that  which  is 
•over  our  heads.     The  apparent  sky  is  not  a  hemisphere,  but  a 
flattened  semi-ellipsoid. 

469.  Magnitude  and  Distance  Associated.— Our  judg- 
ments of  distance  and  of  magnitude  are  closely  associated.    If 
objects  are  known,  we  estimate  their  distance  by  their  visual  angle, 
as  has  been  stated ;  but  if  unknown,  we  must  first  acquire  our 
notion  of  their  distance  by  some  other  means,  and  then  their 
visual  angle  gives  us  a  definite  impression  as  to  their  size.     And  if 
our  judgment  of  distance  is  erroneous,   a  corresponding   error 
attaches  to  our  estimate  of  their  magnitude.     An  insect  crawling 
slowly  on  the  window,  if  by  mistake  it  is  supposed  to  be  some 
rods  beyond  the  window,  will  appear  like  a  bird  flying  in  the  air. 
The  moon  near  the  horizon  seems  larger  than  above  us,  because 
we  are  able  to  locate  it  at  a  greater  distance. 


304  OPTICS. 

The  apparent  linear  dimensions  of  objects  are  directly  propor- 
tional to  their  actual  dimensions  and  inversely  proportional  to 
their  distances  from  the  observer. 

470.  Binocular  Vision.— The  Stereoscope. — If  objects  are 
placed  quite  near  us,  we  obtain  simultaneously  two  views,  which 
are  essentially  different  from  each  other — one  with  one  eye,  and 
one  with  the  other.  By  the  right  eye  more  of  the  right  side,  and 
less  of  the  left  side,  is  seen,  than  by  the  left  eye.  Also,  objects  in 
the  foreground  fall  further  to  the-  left  compared  with  distant  ob- 
jects, when  seen  with  the  right  eye  than  when  seen  with  the  left. 
And  we  associate  with  these  combined  views  the  form  and  extent 
of  a  body,  or  group  of  bodies,  particularly  in  respect  to  distance 
of  parts  from  us.  It  is,  then,  by  means  of  vision  with  two  eyes,  or 
binocular  vision,  that  we  are  enabled  to  get  accurate  perceptions 
of  prominence  or  depression  of  surface,  reckoned  in  the  visual 
direction.  A  picture  offers  no  such  advantage,  since  all  its  parts 
are  on  one  surface,  at  a  common  distance  from  the  eyes.  But,  if 
two  perspective  views  of  an  object  should  be  prepared,  differing 
as  those  views  do,  which  are  seen  by  the  two  eyes,  and  if  the  right 
eye  could  then  see  only  the  right-hand  view,  and  the  left  eye  only 
the  left-hand  view,  and  if,  furthermore,  these  two  views  could  be 
made  to  appear  on  one  and  the  same  ground,  the  vision  would 
then  be  the  same  as  is  obtained  of  the  real  object  by  both  eyes. 
This  is  effected  by  the  stereoscope.  Two  photographic  views  are 
taken,  in  directions  which  make  a  small  angle  with  each  other, 
and  these  views  are  seen  at  once  by  the  two  eyes  respectively, 
through  a  pair  of  half -lenses,  placed  with  their  thin  edges  toward 
each  other,  so  as  to  turn  the  visual  pencils  away  from  each  other, 
as  though  they  emanated  from  one  object.  An  appearance  of 
relief  and  reality  is  thus  given  to  superficial  pictures,  precisely 
like  that  obtained  from  viewing  the  objects  themselves. 


CHAPTER    IX 

OPTICAL    INSTRUMENTS. 

471.  The  Camera  Lucida. — This  is  a  four-sided  prism,  so 
contrived  as  to  form  an  apparent  image  at  a  surface  on  which  that 
image  may  be  copied,  the  surface  and  image  being  both  visible  at 
the  same  time.  It  has  the  form  represented  by  the  section  in 


THE    MICROSCOPE. 


305 


Fig.  284;  A  ==  90°,  C  =  135°;  #  and  D,  of  any  convenient  size, 
their  sum  of  course  =  135°.  A  pencil  of  light  from  the  object 
M,  falling  perpendicularly  on  A  D,  proceeds  on,  and  makes,  with 
D  C,  an  angle  equal  to  the  complement  of  D.  After  suffer- 
ing total  reflection  at  G,  and  again  at 
H,  its  direction  H  E  is  perpendicular 
to  M  F.  For,  produce  M  F,  and  EH, 
till  they  intersect  in  /;  then,  since 
C  =  135°,  C  G  H+  OH  G  =  45°  ;  but 
/  G  H=  2  C  G  H,  and  IE  G  =  2 


FIG.  284. 


.-.  /  =  90°.  Therefore  H  E  emerges 
at  right  angles  to  A  B,  and  is  not  re- 
fracted. Now,  if  the  pupil  of  the 
eye  be  brought  over  the  edge  B,  so 
that,  while  E  H  enters,  there  may 
also  enter  a  pencil  from  the  surface 
at  M'9  then  both  the  surface  M'  and 
the  object  M  will  be  seen  coinciding  with  each  other,  and  the  hand 
may  therefore  sketch  M  on  the  surface  at  M' .  The  reason  for 
two  reflections  of  the  light  is,  that  the  inversion  produced  by  one 
reflection  may  be  restored  by  the  second. 

One  of  the  most  useful  applications  of  the  camera  lucida  is  in 
connection  with  the  compound  microscope,  where  it  is  employed 
in  copying  with  exactness  the  forms  of  natural  objects,  too  small 
to  be  at  all  visible  to  the  naked  eye. 

472.  The  Microscope. — This  is  an  instrument  for  viewing 
minute  objects.     The  nearer  an  object  is  brought  to  the  eye,  the 
larger  is  the  angle  which  it  fills,  and  therefore  the  more  perfect  is 
the  view,  provided  the  rays  of  each  pencil  are  converged  to  a  point 
on  the  retina.     But  if  the  object  is  nearer  than  the  limit  of  dis- 
tinct vision,  the  eye  is  unable  to  produce  sufficient  convergency. 
If  the  letters  of  a  book  are  brough  t  close  to  the  eye,  they  become 
blurred  and  wholly  illegible.    But  let  a  pin-hole  be  pricked  through 
paper,  and  interposed  between  the  eye  and  the  letters,  and,  though 
faint,  they  are  distinct  and  much  enlarged.    The  distinctness  is 
owing  to  the  fact  that  the  outer  rays,  which  are  most  divergent, 
are  excluded,  and  the  eye  is  able  to  converge  the  few  central  rays 
of  each  pencil  to  a  focus.     The  letters  appear  magnified,  because 
they  are  so  near,  and  fill  a  large  angle.    The  microscope  utilizes 
these  excluded  rays,  and  renders  the  image  not  only  large  and  dis- 
tinct, but  luminous. 

473.  The  Single  Microscope. — The  single  microscope  is 
merely  a  convex  lens.     It  aids  the  eye  in  converging  the  rays, 

20 


306 


OPTICS. 


which  come  from  a  very  near  object,  so  that  a  distinct  and  lumi- 
nous image  may  be  formed  on  the  retina. 

Let  P  Q  (Fig.  285)  be  the  object,  and  ;;  q  its  image.     We 


FIG.  285. 


have    j     =  --    5  but> 


383,   C  q  = 


Art- 


which 


The 


. 

visual  angle  of  p  q,  at  the  dis- 


-_- 


of  P  §,  a£  Me 
m  their  ratio  we  have 
pq 


P  0 
distance  of  distinct  vision,  v,  is  -  —  ; 


calling 


~ 


PQ 


F-CQ 


EC+Cq 


F 


X 


~  EC+  Cq       F-  C  Q' 
Making  E  C  =  0,  by  bringing  the  eye  close  to  the  lens,  we  have 

=-    M  °  Q  =  F>  then  m  =      "      If 


.0~q 


=~   X 


- 
maximum  effect  of  the  lens. 


C  q  =^  v,  then  0  Q  = 


=  —  +1,  the 


whence  m  =  — 
Hence,  the  magnifying  power  of  a 

1J  1) 

lens,  the  eye  being  close  to  it,  is  between  the  limits  -=-,  and  -^  -f  1, 

Jc  U 

according  to  the  position  of  the  object. 

Though  the  focal  distance  of  a  lens  may  be  made  as  small  as 
we  please,  yet  a  practical  limit  to  the  magnifying  power  is  very 
soon  reached. 

1.  The  field  of  view,  that  is,  the  extent  of  surface  which  can 
be  seen  at  once,  diminishes  as  the  power  is  increased. 

2.  Spherical  aberration  increases  rapidly,   because  the  outer 
rays  are  very  divergent.     Hence  the  necessity  of  diminishing  the 
aperture  of  the  lens,  in  order  to  exclude  the  most  divergent  rays. 

3.  It  is  more  difficult  to  illuminate  the  object  as  the  focal 
length  of  the  lens  becomes  less  ;  and  this  difficulty  becomes  a 
greater  evil  on  account  of  the  necessity  of  diminishing  the  aper- 
ture in  order  to  reduce  the  spherical  aberration. 

To  lessen  the  spherical  aberration,  two  or  more  plano-convex 
lenses  are  used,  close  together. 

Magnifying  glasses  are  single  microscopes  of  low  power,  such  as 


MODERN    IMPROVEMENTS.  307 

are  used  by  watchmakers.    Lenses  of  still  lower  power  an/  several 
inches  in  diameter  are  used  for  viewing  pictures. 

474.  The  Compound  Microscope. — It  is  so  called  because 
it  consists  of  two  parts,  an  object-glass,  by  which  a  real  and  mag- 
nified image  is  formed,  and  an  eye-glass,  by  which  that  image  is 
again  magnified.    Its  general  principle  may  be  explained  by  Fig. 
286,  in  which  a  b  is  the  small  object,  c  d  the  object- 
glass,  and  ef  the  eye-glass.     Let  a  b  be  a  little 

beyond  the  principal  focus  of  c  d,  and  then  the 

image  g  h  will  be  real,  on  the  opposite  side  of  c  d, 

and  larger  than  a  b.    Now  apply  ef  as  a  single 

microscope  for  viewing  g  h,  as  though  it  were  an 

object  of  comparatively  large  size.     Let  g  li  be  at 

the  principal  focus  of  ef,  so  that  the  rays  of  each 

pencil  shall  be  parallel ;  they  will,  therefore,  come 

to  the  eye  at  k,  from  an  apparent  image  on  the 

same  side  as  the  real  one,  g  h ;  and  the  extreme 

pencils,    e  k,  f  k,  if   produced   backward,    will 

include  the  image  between  them,  e  k  f  being  the  angle  which 

it  fills. 

475.  The  Magnifying  Power. — The  magnifying  power  of 
the  compound   microscope  is    estimated   by   compounding  two 
ratios ;  first,  the  distance  of  the  image  from  the  object-glass,  to 
the  distance  of  the  object  from  the  same  ;  and  secondly,  the  limit 
of  distinct  vision  to  the  distance  of  the  image  from  the  eye-glass. 
For  the  image  itself  is  enlarged  in  the  first  ratio  (Art.  385)  ;  and 
the  eye-glass  enlarges  that  image  in  the  second  ratio  (Art.  473). 
The  advantage  of  this  form  over  the  single  microscope  is  not  so 
much  that  a  great  magnifying  power  is  obtained,  as  that  a  given 
magnifying  power  is  accompanied  by  8.  larger  field  of  view. 

476.  Modern  Improvements. — Great  improvements  have 
been  made  in  the  compound  microscope,  principally  by  combining 
lenses  in  such  a  manner  as  greatly  to  reduce  the  chromatic  and 
spherical  aberrations.     The  object-glass  generally  consists  of  one, 
two,  or  three  achromatic  pairs  of  lenses.     The  eye-piece  usually 
contains  two  plano-convex  lenses,  a  combination  which  is  found 
to  be  the  most  favorable  for  diminishing  the  spherical  aberration, 
and  for  enlarging  the  field  of  view. 

In  Fig.  28?  let  a  ~b  be  the  object,  C  an  achromatic  lens,  called 
the  objective,  D  the  field  lens  (so  called  because  it  enlarges  the 
field  of  view  by  bending  the  pencil  which  would  come  to  a  focus 
at  a',  and  pass  below  the  eye  lens,  so  that  it  may  come  to  a  focus 
at  a"}  and  thence  pass  into  the  eye  lens),  E  the  eye  lens  which 


308 


OPTICS. 


renders  the  rays  of  the  pencil  so  nearly  parallel  that  the  eye  re- 
ceives them  as  though  coming  from  the  point  a'". 


FIG.  287. 


The  method  of  determining  the  magnifying  power  given  in 
the  last  paragraph,  is  not  applicable  in  this  form  of  instrument, 
but  an  experimental  determination  is  made  as  follows:  A  very 
finely-divided  scale  is  placed  under  the  microscope  ;  a  mirror,  from 
which  a  small  part  of  the  silvering  has  been  removed,  is  placed 
near  the  eye-piece  at  an  angle  of  45°  with  the  axis  of  the 
instrument,  and  behind  this  at  the  distance  of  ordinary  distinct 
vision,  about  ten  inches,  and  visible  through  the  unsilvered 
part  of  the  mirror,  is  placed  a  second  scale  like  the  first ;  the 
number  of  divisions  of  the  second  scale  covered  by  one  division 
seen  through  the  microscope  by  reflection  gives  the  power.  Any 
change  in  the  relative  positions  of  the  lenses,  changes  the  magni- 
fying power. 

FIG.  288. 


477.  The  Magic  Lantern. — It  consists  of  a  box,  repre- 
sented in  Fig.  288,  containing  a  lamp,  and  having  openings  so 
arranged  as  to  permit  the  air  to  pass  freely  through  it,  without 
letting  light  escape.  In  front  of  the  lamp  is  a  tube  containing  a 
concentrating  lens,  C,  the  painting  on  glass,  B,  and  the  lens,  A, 
for  producing  the  image  ;  back  of  the  lamp  may  be  a  concave 


THE    ASTRONOMICAL    TELESCOPE. 


309 


mirror  for  reflecting  additional  light  on  the  lens  0.  The  trans- 
parency B  is  a  painting  on  glass,  and  the  strong  light  which  falls 
on  it  proceeds  through  the  lens  A,  as  from  an  original  object 
brilliantly  colored.  It  is  a  little  further  from  A  than  its  princi- 
pal focus,  and  therefore  the  rays  from  any  point  are  converged  to 
the  conjugate  focus  in  a  real  image,  F,  on  a  distant  screen.  This 
image  is  of  course  inverted  relatively  to  the  object,  and  therefore, 
if  the  picture  B  is  inverted,  F  will  be  erect.  The  lens  may  be 
placed  at  various  distances  from  B  by  the  adjusting  screw  a,  so  as 
to  give  the  greatest  distinctness  to  the  image  at  any  given  distance 
of  the  screen.  According  to  Art.  385,  the  diam.  of  B  :  diam.  of 
F : :  A  B  :  A  F\  and  therefore,  theoretically,  the  image  may  be 
as  large  as  we  please. 

478.  The  Telescope. — The  telescope  aids  in  viewing  dis- 
tant bodies.    An  image  of  the  distant  body  is  first  formed  in  the 
principal  focus  of  a  convex  lens  or  a  concave  mirror ;  and  then  a 
microscope  is  employed  to  magnify  that  image  as  though  it  were  a 
small  body.    The  image  is  much  more  luminous  than  that  formed 
in  the  eye,  when  looking  at  the  heavenly  body,  because  there  is 
concentrated  in  the  former  the  large  beam  of  light  which  falls 
upon  the  lens  or  mirror,  while  the  latter  is  formed  by  the  slender 
pencil  only  which  enters  the  pupil  of  the  eye.     If  the  image  in  a 
telescope  is  formed  by  a  lens,  the  instrument  is  called  a  refracting 
telescope  ;  but  if  by  a  mirror,  a  reflecting  telescope. 

479.  The   Astronomical  Telescope. — This  is  the  most 
simple  of  the  refracting  telescopes,  consisting  of  a  lens  to  form  an 
image  of  the  heavenly  body,  and  a  single  microscope  for  magnify- 
ing that  image.     The  former  is  called  the  object-glass,  the  latter 
the  eye-glass. 

Let  a  (Fig.  289)  be  the  image  of  some  point  of  a  heavenly 
body,  the  divergent  rays  from  which,  marked  A  A'  A,  are  practi- 

FIG.  289. 


R 


cally  parallel,  and  b  the  image  of  the  point  B  B'  B.  As  the  rays 
forming  these  images  are  parallel  rays,  a  b  is  at  the  principal 
focus  of  the  object  glass  0.  The  eye  lens  E  receives  the  direr- 


310  OPTICS. 

gent  pencils  from  a  and  J,  bends  them  so  that  they  enter  the  eye 
as  parallel  or  nearly  parallel  beams  coming  apparently  from  the 
direction  of  a'  and  b'.  The  image  a  b  is  situated  at  the  principal 
focus  of  E,  the  distance  between  the  lenses  0  and  E  being  the 
sum  of  their  principal  focal  distances. 

480.  The  Powers  of  the  Telescope. — The  magnifying 
power  of  the  astronomical  telescope  is  expressed  by  the  ratio  of  the 
focal  distance  of  the  object-glass  to  that  of  the  eye-glass.  For  (Fig. 
289)  the  object  as  it  would  be  seen  by  the  eye  if  placed  at  0  fills 
the  angle  A'  0  B'  between  the  axes  of  its  extreme  pencils.  But, 
since  the  axes  cross  each  other  in  straight  lines  at  the  optic  centre 
of  the  lens,  A'  0  B'  =  a  Ob.  Therefore,  to  an  eye  placed  at  the 
object-glass,  the  image,  a  b,  appears  just  as  large  as  the  object ; 
while  at  the  eye-glass  it  appears  as  much  larger  in  diameter  as  the 
distance  is  less. 

Since  no  simple  eye  lenses  are  used,  and  as  the  equivalent 
power  of  the  compound  eye-piece  is  not  readily  found,  the  follow- 
ing practical  method  of  finding  the  power  of  an  astronomical 
telescope  is  of  use  : 

Adjust  the  eye-piece  so  that  a  sharp  and  clear  view  of  some 
very  distant  object  may  be  obtained  ;  remove  the  object-glass,  and 
in  its  place  put  an  opaque  card  disk  in  which  is  cut  an  opening  of 
the  shape  of  a  very  flat  isosceles  triangle,  whose  base  is  nearly  as 
long  as  the  diameter  of  the  object-glass  ;  receive  an  image  of  this 
opening  upon  a  translucent  glass  or  paper  screen  held  close  to  the 
eye-piece,  and  measure  the  base  of  the  image  very  exactly ;  the 
length  of  the  base  of  the  opening  divided  by  the  base  of  its  image 
is  the  power.  If  any  of  the  rays  from  the  opening  are  cut  off  by 
diaphragms  in  the  tube  the  imperfection  of  the  image  will  make 
known  the  difficulty,  which  must  be  removed. 

The  field  of  view  is  determined  thus  :  Direct  the  telescope  to  a 
star  on  or  near  the  celestial  equator,  and  note  the  time  in  seconds 
which  the  star  occupies  in  passing  across  the  diameter  of  the  field 
of  view ;  divide  this  time  by  4  and  the  quotient  will  be  the  diame- 
ter of  the  field  in  minutes  of  arc,  because  a  star  on  the  equator 
moves  through  one  minute  of  arc  in  four  seconds  of  time. 

The  illuminatiny  power  is  important  for  objects  which  shed  a 
very  feeble  light  on  account  of  their  immense  distance.  This 
power  depends  on  the  size  of  the  beam,  that  is,  on  the  aperture  of 
the  object-glass. 

The  defining  power  is  the  power  of  giving  a  clear  and  sharply- 
defined  image,  without  which  both  the  other  powers  are  useless. 
And  it  is  the  power  of  producing  a  well-defined  image  which 
limits  both  of  the  other  powers.  For  every  attempt  to  increase 


THE  TERRESTRIAL  TELESCOPE. 


311 


the  magnifying  power  by  giving  a  large  ratio  to  the  focal  lengths 
of  the  object-glass  and  the  eye-glass,  or  to  increase  the  illuminat- 
ing power  by  enlarging  the  object-glass,  increases  the  difficulties 
in  the  way  of  getting  a  perfect  image.  These  difficulties  are  three 
— the  spherical  aberration  (Art.  388),  the  chromatic  aberration 
(Art.  402),  and  unequal  densities  in  the  glass.  The  third  diffi- 
culty is  a  very  serious  one,  especially  in  large  lenses. 

481.  The  Terrestrial  Telescope. — In  order  to  secure  sim- 
plicity, and  thus  the  highest  excellence,  in  the  astronomical  tele- 
scope, the  image  is  allowed  to  be  inverted,  which  circumstance  is 
of  no  importance  in  viewing  heavenly  bodies.  But,  for  terrestrial 
objects,  it  would  be  a  serious  inconvenience ;  and,  therefore,  a  ter- 
restrial telescope,  or  spyglass,  has  additional  lenses  for  the  purpose 
of  forming  a  second  image,  inverted,  compared  with  the  first,  and, 
therefore,  erect,  compared  with  the  object.  In  Fig.  290,  m  m  m 
represent  a  pencil  of  rays  from  the  top  of  a  distant  object,  and 

FIG.  290. 


n  n  n  from  the  bottom ;  A  B,  the  object-glass ;  m  n,  the  first 
image;  CD,  the  first  eye-glass,  which  converges  the  pencils  of 
parallel  rays  to  L.  Instead  of  placing  the  eye  at  L,  the  pencils 
are  allowed  to  cross  and  fall  on  the  second  eye-glass,  E  F,  by  which 
the  rays  of  each  pencil  are  converged  to  a  point  in  the  second 
image,  m'  n',  which  is  viewed  by  the  third  eye-glass,  G  H.  The 
second  and  third  lenses  are  commonly  of  equal  focal  length,  and 
add  nothing  to  the  magnifying  power. 

Such  instruments  are  usually  of  a  portable  size,  and  hence  the 
aberrations  are  corrected  with  comparative  ease,  by  the  methods 
already  described.  The  spy-glass,  for  convenient  transportation, 
is  made  of  a  series  of  tubes,  which  slide  together  in  a  very  com- 
pact form. 

If  the  lenses  C  D  and  E  F  are  of  the  same  power  they  do  not 
affect  the  power  of  the  telescope,  which  may  then  be  represented 

Tjl 

as  in  the  astronomical  telescope  by  -^-. 

To  determine  the  power  practically,  look  at  some  distant  scale 
of  equal  parts,  a  brick  wall  for  instance,  and  keeping  both  eyes 


312  dPTICS. 

open,  note  how  many  bricks  as  seen  by  the  unaided  eye  are  covered 
1/y  the  image  of  one  brick  as  seen  through  the  telescope  ;  this 
number  so  covered  is  the  expression  for  the  power.  If  one  space, 
for  instance,  seen  through  the  telescope,  covers  twenty  spaces  seen 
with  the  unaided  eye,  the  telescope  magnifies  twenty  diameters. 

482.  Galileo's  Telescope.— This  was  the  first  form  of 
telescope,  having  been  invented  by  Galileo,  whose  name  it  there- 
fore bears.  It  differs  from  the  common  astronomical  telescope  in 
having  for  the  eye-glass  a  concave  instead  of  a  convex  lens,  which 
receives  the  rays  at  such  a  distance  from  the  focus  to  which  they 
tend,  as  to  render  them  parallel. 

Thus  the  rays  mm  m  (Fig.  291),  from  a  point  at  the  top  of 
the  object,  are  converged  by  the  object-glass  0  towards  a  focus  a; 

FIG.  291. 


but  before  meeting  at  a  they  fall  upon  a  concave  eye-lens  E,  and 
are  rendered  parallel  or  slightly  divergent,  as  though  they  came 
from  a  point  in  the  direction  indicated  by  M.  The  point  from 
which  the  rays  m  m  m  proceeded,  and  its  virtual  image  M9  are 
both  on  the  same  side  of  the  axis  of  the  instrument,  and  there  is 
no  inversion. 

It  is  obvious  that,  since  the  pencils  diverge,  only  the  central 
ones,  within  the  size  of  the  pupil,  can  enter  the  eye.  This  cir- 
cumstance exceedingly  limits  the  field  of  view,  and  unfits  the 
instrument  for  telescopic  use.  It  is  employed  for  opera-glasses, 
having  a  power  usually  of  only  two  or  three  diameters. 

ET 

The  expression  for  the  power  is  ~,  as  in  the  preceding  forms. 

The  power  may  be  very  readily  determined  practically  as  in  the 
case  of  the  terrestrial  telescope. 

483.  The  Gregorian  Telescope. — This  is  the  most  fre- 
quent form  of  reflecting  telescope,  and  receives  its  name  from  the 
inventor,  Dr.  Gregory,  of  Scotland. 

Let  A  (Fig  292)  be  a  point  of  a  very  distant  body  from  which 
rays,  practically  parallel,  fall  upon  the  large  concave  mirror  M, 
which  is  perforated  through  the  middle  o  o' ;  these  are  converged 
to  the  principal  focus  a,  and  passing  this  point,  diverging  again, 
are  received  by  the  small  concave  reflector  R  of  short  focus,  and 


THE    NEWTONIAN    TELESCOPE. 


313 


are  made  to  converge  to  a',  forming  a  real  image  ;  thence  the  rays 
diverge  once  more,  and  falling  upon  the  eye-glass  E,  are  refracted 
as  though  they  came  from  an  object  in  the  direction  A'. 

The   Cassegrainian   telescope  differs  from  the  Gregorian  in 
having  a  convex  reflector  in  place  of  the  concave  R;  this  is  so 


FIG.  292. 


placed  as  to  receive  the  rays  before  they  reach  the  focus  a,  and, 
by  rendering  them  less  convergent,  bring  them  to  a  focus  at  th& 
place  of  the  image  «',  but  upon  the  same  side  of  the  axis  as  a. 

484.  The  Newtonian  Telescope. — R  is  a  concave  reflec- 
tor (Fig.  293),  F  a  plane  mirror  called  a  flat,  E  the  convex  eye- 
glass. Rays  from  some  point  A  of  a  distant  object  are  converged 
by  R  towards  the  principal  focus  a ;  they  are  intercepted  by  the 
jlat  F  and  turned  aside,  without  change  of  convergency,  to  the 


FIG. 


focus  a',  passing  which  they  fall  upon  the  eye-lens  E,  and  enter 
the  eye  as  though  they  came  from  the  direction  A'.  The  magni- 
fying power  = 

focal  length  of  reflector 

focal  length  of  eye-glass ' 

485.  The  Herschelian  Telescope.— Sir  William  Herschel 
modified  the  Newtonian  by  dispensing  with  the  small  reflector  F, 
and  inclining  the  large  speculum  R,  so  as  to  form  the  image  near 
the  edge  of  the  tube,  where  the  eye-glass  is  attached.  Thus,  the 


314 


OPTICS. 


observer  is  situated  with  his  back  to  the  object.  The  speculum  of 
Herschel's  telescope  was  about  four  feet  in  diameter,  and  weighed 
more  than  2,000  pounds,  and  its  focal  length  was  forty  feet.  The 
Earl  of  Rosse  has  since  constructed  a  Herschelian  telescope  having 
an  aperture  of  six  feet,  and  a  focal  length  of  fifty  feet.  The 
magnifying  power  is  the  same  as  in  the  Newtonian. 

486.  Eye-pieces,  or  Oculars. — The  negative,  or  Huyghe- 
nian,  eye-piece  consists  of  two  plano-convex  lenses  of  crown  glass, 
jPand  E  (Fig.  294),  the  convex  surfaces  being  turned  towards  the 
object-glass.  A  pencil  of  rays  from  the  object-glass,  converging 

to  a  principal  focus  «,  is  bent 

FlG  29i  from  its  course   by  F  and 

brought  to  a  focus  a',  half 
way  between  the  two  lenses. 
The  image  found  at  a'  is 
then  viewed  by  the  eye-lens 
E  as  usual. 

This  eye-piece  is  called  negative  because  it  is  adapted  to  rays 
already  converging.  The  focal  length  of  Fis  three  times  that  of 
E,  and  the  distance  between  the  lenses  is  one-half  the  sum  of  the 
focal  lengths.  This  combination  is  achromatic,  for  the  following 
reasons :  the  deviation  of  a  ray  by  a  convex  lens  is  greater,  the 
greater  the  distance  of  the  point  of  incidence  upon  the  lens  is 
from  the  axis  of  the  lens ;  a  ray  refracted  by  F  is  separated  into 
its  component  colors,  the  red  ray  being  least  bent  towards  the 
centre  of  the  eye-lens  E  and  the  violet  most ;  the  red  ray  falls 
upon  E  at  a  greater  distance  from  the  centre  than  the  violet,  and 
being  more  bent  from  its  course  than  the  violet,  they  emerge 
parallel. 

The  positive,  or  Ramsden,  eye-piece  consists  of  two  plano- 
convex  lenses  E'  and  E  (Fig.   295)  with  the  convex  surfaces 
turned  toward  each  other.     The  rays  from  the  object-glass  are 
focussed  at  a  and  thence 
pass  to  the  eye  as  indi- 
cated in  the  figure.    Two 
lenses  are  used   instead 
of  one    in    order   more 
easily  and    perfectly   to 

correct  spherical  aberration.  E'  and  E  are  of  equal  focal  lengths, 
and  the  distance  between  them  is  two-thirds  the  focal  length  of 
one  of  them.  This  combination  is  not  achromatic.  It  is  always 
used  when  spider  lines  are  placed  in  the  focus  of  the  object-glass 
for  purposes  of  exact  measurement 


PART    VI. 


CHAPTER  I. 

EXPANSION  BY  HEAT.— THE    THERMOMETER. 

487.  Nature  of  Heat.— There  is  abundant  reason  for  be- 
lieving that  heat  consists  of  exceedingly  minute  and  rapid  vibra- 
tions of  ordinary  matter  and  of  the  ether  which  tills  all  space.     It 
is  to  be  regarded  as  one  of  the  modes  of  motion,  which  may  be 
caused  by  any  kind  of  force,  and  which  may  be  made  a  measure 
of  that  force.     Heat  affects  only  one  of  our  senses,  that  of  feeling. 
Its  increase  produces  the  sensation  of  warmth,  and  its  diminution 
that  of  cold. 

488.  Expansion  and  Contraction  by  Heat  and  Cold. 

— It  is  found  to  be  a  fact  almost  without  exception,  that  as  bodies 
are  heated  they  are  expanded,  and  that  they  contract  as  they  are 
cooled.  It  is  easy  to  conceive  that  the  vibratory  motion  of  the 
several  molecules  of  a  body  compels  them  to  recede  from  each 
other,  and  fco  recede  the  more  as  the  vibration  becomes  more 
violent.  Although  the  change  in  magnitude  is  generally  very 
small,  yet  it  is  rendered  visible  by  special  contrivances,  and  is 
made  the  means  of  measuring  temperature. 

489.  Expansion    of  Solids. — When  the  expansion  of  a 
solid  is  considered  simply  in  one  dimension,  it  is  called  linear 
expansion ;  in  two  dimensions  only,  superficial  expansion  ;  in  all 
three  dimensions,  cubical  expansion. 

The  linear  expansion  of  a  metallic  rod  is  readily  made  visible 
by  an  instrument  called  the  pyrometer,  which  magnifies  the  mo- 
tion. The  end  A  of  the  rod  A  B  (Fig.  296)  is  held  in  place  by 
a  screw.  The  end  B  rests  against  the  short  arm  of  the  lever  (7,  the 
longer  arm  of  which  bears  on  the  arm  D  of  the  long  bent  lever 
D  E ;  this  serves  as  an  index  to  the  graduated  arc  E  F.  The  long 
metallic  dish  G  G,  being  raised  on  the  hinges  H  H,  so  as  to 


316  HEAT. 

enclose  the  bar  A  B,  and  then  filled  with  hot  water,  the  oar 
instantly  expands,  and  raises  the  index  along  the  arc  E  F. 

FIG.  296. 


490.  Coefficient  of  Expansion. — The  coefficient  of  linear 
expansion  of  a  given  substance  is  the  fractional  increase  of  its 
length,  when  its  temperature  is  raised  one  degree.     But  since  this 
increase  is  generally  somewhat  greater  at  higher  temperatures,  the 
coefficients  of  expansion  given  in  tables  usually  refer  to  a  tempera- 
ture at  or  near  the  freezing  point  of  water.     Thus  the  coefficient 
of  expansion  for  silver  is  0.000019097  ;  by  which  is  meant  that  a 
silver  bar  one  foot  long  at  0°  C.  becomes  1.000019097  ft.  in  length 
at  1°  C. 

The  coefficient  of  superficial  expansion  is  twice,  and  that  of 
cubical  expansion  three  times  as  great  as  the  coefficient  of  linear 
expansion.  For,  suppose  c  to  be  the  coefficient  of  linear  expan- 
sion ;  then  if  the  edge  of  a  cube  is  1,  and  the  temperature  is  raised 
1°,  the  edge  becomes  1  +  c,  and  the  area  of  one  side  becomes 
(1  +  c)*  =  l  +  2c+  c*,and  the  volume  (1  +  c)3  =  l  +  3c  +3c?-f  c3. 
But  as  c  is  very  small,  the  higher  powers  may  be  neglected,  and 
the  area  is  l  +  2c,  and  the  volume  is  J  +  3c ;  that  is,  the  coefficient 
of  superficial  expansion  is  2c,  and  that  of  cubical  expansion  is  3c, 
as  stated  above. 

491.  The  Coefficient  of  Expansion  differs  in  different 
Substances. — Copper  expands  nearly  twice  as  much  as  platinum 
for  a  given  increase  of  temperature ;  the  ratio  of  expansion  in 
steel  and  brass  is  about  as  61  to  100.     This  ratio  is  employed  in 
the  construction  of  the  compensation  pendulum  (Art.  164). 

If  two  thin  slips  of  metal  of  different  expansibility  be  soldered 
together  so  as  to  make  a  slip  of  double  thickness,  it  will  bend  one 
way  and  the  other  by  changes  of  temperature.  If  it  is  straight 
at  a  certain  temperature,  heating  will  bend  it  so  as  to  bring  the 
most  expansible  metal  on  the  convex  side  ;  and  cooling  will  bend 
it  in  the  opposite  direction  ;  and  the  degree  of  flexure  will  be  ac- 
cording to  the  degree  of  change  in  temperature.  Compensation 
in  clocks  and  watches  is  sometimes  effected  on  this  plan.  If  the 


THE    THERMAL    FORCE.  317 

compound  slip  has  the  form  of  a  helix,  with  the  most  expansible 
metal  on  the  inside,  heating  will  begin  to  uncoil  it,  and  cooling, 
to  coil  it  closer.  A  very  sensitive  thermometer,  known  as  Bre- 
guet's  thermometer,  is  constructed  on  this  principle. 

As  notable  exceptions  to  the  general  rule  that  solids  expand 
when  heated,  may  be  mentioned  stretched  India-rubber,  and  also 
Rose's  fusible  metal,  an  alloy  of  2  parts  bismuth,  1  part  lead,  and 
1  part  tin ;  the  latter  compound  expands  up  to  44°  C.,  then 
rapidly  contracts  up  to  69°  C.,  which  is  the  temperature  of  maxi- 
mum density,  and  again  expands  till  it  melts  at  94°  C. 

492.  The  Strength  of  the  Thermal  Force.— It  is  found 
that  the  force  exerted  by  a  body,  when  expanding  by  heat  or  con- 
tracting by  cold,  is  equal  to  the  mechanical  force  necessary  to 
expand  or  compress  the  body  to  the  same  degree.     The  force  is 
therefore  very  great.     If  the  rails  were  to  be  fitted  tightly  end  to 
end  on  a  railroad,  they  would  be  forced  out  of  their  places  by 
expansion  in  warm  weather,  and  the  track  ruined.     The  tire  of  a 
carriage  wheel  is  heated  till  it  is  too  large,  and  then  put  upon  the 
wheel ;  when  cool,  it  draws  together  the  several  parts  with  great 
firmness.     In  repeated  instances,  the  walls  of  a  building,  when 
they  have  begun  to  spread  by  the  lateral  pressure  of  an  arched 
roof,  have  been  drawn  together  by  the  force  of  contraction  in  cool- 
ing.   A  series  of  iron  rods  being  passed  across  the  building  through 
the  upper  part  of  the  walls,  and  broad  nuts  being  screwed  upon 
the  ends,  the  alternate  bars  are  expanded  by  the  heat  of  lamps, 
and  the  nuts  tightened.     Then,  when  they  cool,  they  draw  the 
walls  toward  each  other.     The  remaining  bars  are  then  treated  in 
the  same  manner,  and  the  process  is  repeated  till  the  walls  are 
restored  to  their  vertical  position  and  secured. 

493.  Expansion  of  Liquids. — As  liquids  have  no  perma- 
nent form,   the   coefficient    of    expansion    for    them  is  always 
understood  to  be  that  of  cubical  expansion.     There  is  a  practical 
difficulty  in  the  way  of  finding  the  coefficient  for  liquids,  because 
they  must  be  enclosed  in  some  solid,  which  also  expands  by  heat. 
Hence,  the  apparent  expansion  must  be  corrected  by  allowing  for 
the  expansion  of  the  inclosing  solid,   before   the   coefficient  of 
absolute  expansion  is  known. 

This  fact  is  illustrated  by  the  following  experiment.  Fill  the 
bulb  and  part  of  the  stem  of  a  large  thermometer  tube  with  a 
colored  liquid,  and  then  plunge  the  bulb  quickly  into  hot  water ; 
the  first  effect  is,  that  the  liquid  falls,  as  if  it  were  cooled  ;  after  a 
moment  it  begins  to  rise,  and  continues  to  do  so  till  it  attains  the 
temperature  of  the  hot  water.  The  first  movement  is  caused  by 
the  expansion  of  the  glass,  which  is  heated  so  as  to  enlarge  its 


318  HEAT. 

capacity  and  let  down  the  liquid  before  the  heat  has  penetrated 
the  latter.  It  is  obvious  that  what  is  rendered  visible  in  this  case, 
must  always  be  true  when  a  liquid  is  heated — namely,  that  the 
vessel  itself  is  enlarged,  and  therefore  that  the  rise  of  the  liquid 
shows  only  the  difference  of  the  two  expansions.  Ingenious 
methods  have  been  devised  for  obtaining  the  coefficients  of  ab- 
solute expansion  of  liquids,  and  the  results  are  to  be  found  in 
tables  on  this  subject. 

From  the  examination  of  such  tables  we  learn  :  (1)  That 
liquids  expand  more  than  solids  for  a  given  increase  of  tem- 
perature ;  (2)  that  the  coefficient  of  expansion  increases  with  the 
rise  of  temperature ;  (3)  that  the  more  volatile  the  liquid  the  more 
rapidly  will  it  expand  for  a  given  rise  of  temperature. 

494.  Exceptional  Case. — There  is  a  very  important  excep- 
tion to  the  general  law  of  expansion  by  heat  and  contraction  by 
cold,  in  the  case  of  water  just  above  the  freezing  point.     If  water 
be  cooled  down  from  its  boiling  point,  it  continually  contracts  till 
it  reaches  39.1°  F.  or  3.94°  C.,  when  it  begins  to  expand,  and 
continues  to  expand  till  it  freezes  at  32°  F.  or  0°  C.     On  the 
other  hand,  if  water  at  32°  F.  be  heated,  it  contracts  till  it  reaches 
39.1°  F.  or  3.94°  C.,  when  it  commences  to  expand.     Therefore 
the  density  of  water  is  greatest  at  the  point  where  this  change 
occurs.     Different  experimenters  vary  a  little  as  to  its  exact  place, 
but  it  is  usually  called  4°  C..  or  39°  F. 

The  importance  of  this  exception  is  seen  in  the  fact  that  ice 
forms  on  the  surface  of  water,  and  continues  to  float  until  it  is 
again  dissolved.  As  the  cold  of  winter  comes  on,  the  upper 
stratum  of  a  lake  grows  more  dense  and  sinks  ;  and  this  process 
continues  till  the  temperature  of  the  surface  reaches  39°  F.,  when 
it  is  arrested.  Below  that  temperature  the  surface  grows  lighter 
as  it  becomes  colder,  till  ice  is  formed,  which  shields  the  water 
beneath  from  the  severe  cold  of  the  air  above. 

As  in  solids  so  in  liquids,  the  thermal  force  is  very  great. 
Suppose  mercury  to  be  expanded  by  raising  its  temperature  one 
degree,  it  would  require  more  than  300  pounds  to  the  square  inch 
to  compress  it  to  its  former  volume. 

495.  Expansion  of  Gases. — The  gases  expand  by  heat 
more  rapidly  and  more  regularly  than  solids  and  liquids.     The 
large  expansion  and  contraction  of  air  is  made  visible  by  immers- 
ing the  open  end  of  a  large  thermometer  tube  in  colored  liquid. 
When  the  bulb  is  warmed,  bubbles  of  air  are  forced  out  and  rise  to 
the  top  of  the  liquid  ;  when  it  is  cooled,  the  air  contracts  and  the 
liquid  rises  rapidly  in  the  tube. 

Gases,  at  a  constant  pressure,  expand  much  more  than  liquid^ 


THE     THERMOMETER.  319 

or  solids  for  a  given  increment  of  temperature.  All  gases,  at 
temperatures  much  above  that  of  liquefaction,  have  almost  exactly 
the  same  coefficients  of  expansion.  The  coefficient  of  expansion 
for  air  is  ^  fnra  0°  C.  to  1°  C.,  or  ^  from  32°  F.  to  33°  F. 
This  coefficient  increases  slightly  with  increase  of  temperature 
and  pressure. 

To  find  the  volume  of  any  gas  at  0°  C.,  let  v  be  the  known 
volume  at  t°  C.,  also  let  v'  be  the  required  volume  at  0°  C.,  then 


from  which  we  have    v'  =  -  --  ..  —  -.. 

1  T 


If  there  is  a  change  of  pressure,  then,  since  the  tensions  or 
pressures  are  inversely  as  the  volumes,  the  temperatures  being  the 
same  (Art.  237),  have 

v"  :»'::  p'  :  p, 

in  which  v"  is  the  volume  at  0°  C.  and  barometric  pressure  of  30 
inches,  v'  the  volume  at  pressure  p'9  and  p  the  normal  pressure  30 
inches  ;  from  which  we  get,  by  substituting  for  v'  its  value  above, 


x 
V 

496.  The  Thermometer.  —  This  instrument  measures  the 
degree  of  heat,  or  the  temperature,  of  the  medium  around  it,  by 
the  expansion  and  contraction  of  some  substance.  The  substance 
commonly  employed  is  mercury.  The  liquid,  being  inclosed  in  a 
glass  bulb,  can  expand  only  by  rising  in  the  fine  bore  of  the 
stem,  where  very  small  changes  of  volume  are  rendered  visible. 
A  scale  is  attached  to  the  stem  for  reading  the  degrees  of  tem- 
perature. 

The  graduation  of  the  thermometer  must  begin  with  the  fixing 
of  two  important  points  by  natural  phenomena,  the  melting  of  ice 
and  boiling  of  water.  When  the  bulb  is  plunged  into  powdered 
ice,  the  point  at  which  the  column  settles  is  the  freezing  point 
of  the  thermometer.  And  if  it  is  placed  in  steam  under  the 
mean  atmospheric  pressure,  the  mercury  indicates  the  boiling 
point.  Between  these  two  points,  namely  32°  and  212°  F.,  there 
must  be  180°,  and  the  scale  is  graduated  accordingly.  As  the 
bor.e  of  the  tube  is  not  likely  to  be  exactly  equal  in  all  parts,  the 
length  of  the  degrees  should  vary  inversely  as  the  area  of  the 
cross-section.  The  necessary  correction  is  determined  by  moving 
a  short  column  of  mercury  along  the  different  parts  and  com- 
paring the  lengths  occupied  by  it.  The  degrees  in  the  several 
parts  must  vary  in  the  ratio  of  these  lengths. 

The  zero  of  the  scale  tends  to  rise  for  some  time  after  the 
thermometer  is  made,  the  change  amounting  to  more  than  2°  in 


330  HEAT. 

some  instances,  and  therefore  the  instrument  should  not  be  used 
for  at  least  six  months  after  construction.  The  zero  may  also  be 
displaced  by  subjecting  the  instrument  to  high  temperatures. 

497.  Different  Systems   of  Graduation. — There  are  in 
use  three  kinds  of  thermometer  scale,  Fahrenheit's,  Reaumur's, 
and  the  Centigrade  or  Celsius.      In  Fahrenheit's,  the  freezing 
point  of  water  is  called  32°,  and  the  boiling  point,  212° ;  in  Reau- 
mur's, the  freezing  point  is  called  0°,  and  the  boiling  point  80° ; 
in  the  Centigrade,  the  freezing  point  0°,  and  the  boiling  point 
100°.     In  a  scientific  point  of  view,  the  Centigrade  is  preferable 
to  either  of  the  others,  but  Fahrenheit's  is  generally  used  in  this 
country.     The  letter  F.,  R.,  or  C.,  appended  to  a  number  of  de- 
grees, indicates  the  scale  intended.     In  this  country,  F.  is  under- 
stood if  no  letter  is  used. 

498.  To  Reduce  from  one  Scale  to  Another. — Since 
the  zero  of  Fahrenheit's  scale  is  32°  below  the  freezing  point,  while 
in  both  of  the  others  it  is  at  the  freezing  point,  32°  must  always 
be  subtracted  from  any  temperature  according  to  Fahrenheit,  in 
order  to  find  its  relation  to  the  zero  of  the  other  scales.     Then, 
since  212°  —  32°  (=  180°)  F.  are  equal  to  80°  R.,  and  to  100°  C., 
the  formula  for  changing  F.  to  R.  is  $  (F.  —  32)  =  R.  ;  and  for 
changing  F.  to  C.,  it  is  -j-  (F.  —  32)  =  C.    Hence,  to  change  R. 
to  F.,  we  have  £  R.  -f-  32  =  F.  ;   and  to  change  C  to  F.,  £  C. 
+  32  =  F. 

Mercury  congeals  at  about  —38.8°  C.;  therefore,  for  tempera- 
tures lower  than  that,  alcohol  is  used,  which  does  not  congeal 
at  any  known  temperature. 

Above  100°  C.  the  indications  of  the  mercurial  thermometer 
are  not  exact. 

499.  Absolute  Zero  of  Temperature.— At  a  tempera- 
ture of  273°  C.  the  volume  of  a  gas  is  double  its  volume  at  0°  C. 
(Art.  495).     Suppose  that  instead  of  raising  the  temperature,  we 
lower  it ;  for  a  fall  from  0°  to  —1°  C.  the  volume  contracts  ^-j, 
and  for  a  fall  of  273°  it  must  contract  f^-| ;  that  is  to  say,  the 
volume  would  disappear  entirely.     That  the  contraction  would  go 
on  to  —273°  C.  is  not  asserted  ;  but  on  the  supposition  that  the  law 
of  contraction  would  hold,  we  fix  the  temperature  —273°  as  that  at 
which  all  vibrations  would  cease,  and  at  which  consequently  there 
could  be  no  heat  whatever.     The  absolute  zero  more  exactly  given 
is  —273.7°  C.,  and  —460.66°  F.  The  absolute  temperature  is  found 
by  adding  these  readings,  with  signs  changed,  to  the  respective 
readings  of  the  mercurial  thermometer.    As  both  Fahrenheit  and 


CONDUCTION    OP    HEAT    BY    SOLIDS. 


321 


Centigrade  thermometers  are  in  use,  both  will  be  referred  to  in 
the  text,  as  indicated,  that  the  student  may  become  familar  with 
both  systems  of  graduation. 


CHAPTER    II. 


PASSAGE    OF    HEAT    THROUGH    MATTER    AND    SPACE. 


600.  Heat  is  Communicated  in  Several  Ways. — 

1.  By  conduction.    This  is  the  slow  progress  of  the  vibratory 
motion  from  places  of  higher  to  places  of  lower  temperature  in 
the  same  body. 

2.  By  convection.    This  mode  of  communication  takes  placa 
only  in  fluids.    When  the  particles  are  expanded  by  heat,  they  are 
pressed  upward  by  others  which  are  colder  and  therefore  specifi- 
cally heavier.     Heat  is  thus  conveyed  from  place  to  place  by  the 
motion  of  the  heated  matter,  though  the  ultimate  transfer  of  heat 
may  still  take  place  by  conduction. 

3.  By  radiation.     Heat  is  said  to  be  radiated  when  the  vibra- 
tory motion  is  transmitted  from  the  source  with  great  swiftness 
through  the  ether  which  fills  space.    Its  velocity  is  the  same  as 
that  of  light.    The  motion  is  propagated  in  straight  lines  in  every 
direction,  and  each  line  is  called  a  ray  of  heat.     We  feel  the  rays 
of  heat  from  the  sun  or  a  fire,  when  no  object  intervenes  between 
it  and  ourselves. 

501.  Conduction  of  Heat  by  Solids.— Conducted  heat 
passes  through  bodies  very  slowly,  and  yet  at  very  different  rates 
in  different  bodies.  Those  in  which  heat  is  conducted  most 
rapidly,  are  called  good  conductors,  as  the  common  metals  ;  those 
in  which  it  passes  slowly,  are  called  poor  conductors,  as  glass  and 
wood.  In  general,  the  bodies  which  are  good  conductors  of  heat, 
are  also  good  conductors  of  electricity  ;  thus  calling  the  conduc- 
tivity for  electricity  E,  and  for  heat  H,  and  using  silver  as  the 
standard,  we  find — 

Silver E  =100  H  —  100 

Copper "73  "     74 

Gold "    59  "      53 

Brass "22  "      24 

Let  rods  of  different  metals  and  other  substances,  A,  B,  C9 


Iron 

Lead 

German  siiver. 

Bismuth.. 


=13     H  =  12 


11 
6 
2 


322 


HEAT. 


FIG.  297. 


&c.  (Fig.  297),  all  of  the  same  length,  be  inserted  with  water- 
tight joints  in  the  side  of  a  wooden  vessel.  Then  attach  by  wax  a 
marble  under  the  end  of  each  rod,  and  fill  the  vessel  with  boiling 

water.  The  marbles  will  fall 
by  the  melting  of  the  wax,  not 
at  the  same,  but  at  different 
times,  sli owing  that  the  heat 
reaches-  some  of  them  sooner 
than  others.  It  will  be  seen, 
however,  in  the  chapter  on 
specific  heat,  that  the  order  in 
which  they  full  is  not  neces- 
sarily the  order  of  conducting 
power. 

The  amount  of  heat  conducted  through  a  thin  lamina  is 
directly  proportional  to  the  area,  to  the  time  during  which  it 
flows,  to  the  difference  of  temperature  at  the  two  surfaces,  and  to 
the  conductivity  of  the  substance ;  and  is  inversely  proportional 
to  the  thickness  of  the  lamina. 

602.  Effects  of  Molecular  Arrangement.— Organic  sub- 
stances usually  conduct  heat  poorly  ;  and  bodies  having  a  struc- 
tural arrangement  which  differs  in  different  directions,  are  not 
likely  to  conduct  equally  well  in  all  directions.  Thus,  let  two 
thin  plates  be  cut  from  the  same  crystal,  one,  A  (Fig.  298),  per- 

FIG.  298. 


pendicular,  and  the  other,  B,  parallel  to  the  optic  axis.  Let  a 
hole  be  drilled  through  the  centre  of  each,  and  after  a  lamina  of 
wax  has  been  spread  over  the  crystal,  let  a  hot  wire  be  inserted  in 
it.  On  the  plate  A,  the  melting  of  the  wax  will  advance  in  a 
circle,  showing  equal  conducting  power  in  all  directions  in  the 
transverse  section.  In  the  plate  B,  it  Will  advance  in  an  elliptical 
form,  the  major  axis  being  parallel  to  the  optic  axis  of  the  crystal, 
proving  the  best  conduction  to  be  in  that  direction. 

A  block  of  wood  cut  from  one  side  of  the  trunk  of  a  tree,  con- 
ducts most  perfectly  in  the  direction  of  the  fibre,  and  least  in  a 
direction  which  is  tangent  to  the  annual  rings  and  perpendicular 
to  the  fibre,  and  in  an  intermediate  degree  in  the  direction  of  the 
radius  of  the  rings. 


DIFFERENCE    IN    CONDUCTIVE    POWER.        323 

503.  Conduction  by  Fluids. — Fluids,  both  liquid  and  gas- 
eous, are  in  general  very  poor  conductors.     Water,  for  example, 
can  be  made  to  boil  at  the  top  of  a  vessel,  while  a  cake  of  ice  is 
fastened  within  it  a  few  inches  below  the  surface.     If  thermom- 
eters are  placed  at  different  depths,  while  the  water  boils  at  the 
top,  there  is  discovered  to  be  a  very  slight  conduction  of  heat  down- 
ward.     The  gases  conduct  even  more  imperfectly  than  liquids. 

It  will  be  seen  hereafter  (Art.  505)  that  amass  of  fluid  becomes 
heated  by  convection,  not  by  conduction. 

504.  Illustrations  of  Difference  in  Conductive  Power. 

— In  a  room  where  all  articles  are  of  equal  temperature,  some  feel 
much  colder  than  others,  simply  because  they  conduct  the  heat 
from  the  hand  more  rapidly ;  painted  wood  feels  colder  than 
woolen  cloth,  and  marble  colder  still.  If  the  temperature  were 
higher  than  that,  of  the  blood,  then  the  marble  would  seem  the 
hottest,  and  the  cloth  the  coolest,  because  of  the  same  difference 
of  conduction  to  the  hand. 

Our  clothing  does  not  impart  warmth  to  us,  but,  by  its  non- 
conducting property,  prevents  the  vital  warmth  from  being  wasted 
by  radiation  or  conduction.  If  the  air  were  hotter  than  our  blood, 
the  same  clothing  would  serve  to  keep  us  cool. 

A  pitcher  of  water  can  be  kept  cool  much  longer  in  a  hot  day, 
if  wrapped  in  a  few  thicknesses  of  cloth  ;  for  these  prevent  the 
heat  of  the  air  from  being  conducted  to  the  water.  In  the  same 
way  ice  may  be  prevented  from  melting  rapidly. 

The  vibrations  of  heat,  like  those  of  sound,  are  greatly  inter- 
rupted in  their  progress  by  want  .of  continuity  in  the  material. 
Any  substance  is  rendered  a  much  poorer  conductor  by  being  in 
the  condition  of  a  powder  or  fibre.  Ashes,  sand,  sawdust,  wool, 
fur,  hair,  &c.,  owe  much  of  their  non-conducting  quality  to  the 
innumerable  surfaces  which  heat  must  meet  with  in  being  trans- 
mitted through  them. 

Davy's  safety  lamp  is  a  practical  application  of  conduction. 
A  wire  gauze  surrounds  the  lamp,  and  the  air  which  supplies  the 
flame  with  oxygen  can  only  reach  it  by  passing  through  the  gauze. 
A  naked  flame  would  ignite  the  fire  damp  of  the  mines ;  but 
though  the  fire  damp  may  ignite  after  passing  the  gauze  and  may 
fill  the  whole  lamp  with  a  body  of  flame,  yet,  owing  to  the  cooling 
effected  by  the  conduction  of  the  wires,  the  gases  on  the  outside 
are  not  raised  to  the  temperature  of  ignition  ;  thus  warning,  and 
time  for  escape  from  danger,  are  given. 

505.  Convection  of  Heat. — Liquids  and  gases  are  heated 
almost  entirely  by  convection.     As  heat  is  applied  to  the  sides 
and  bottom  of  a  vessel  of  water,  the  heated  particles  become 


324 


HEAT. 


FIG.  299. 


specifically  lighter,  and  are  crowded  up  by  heavier  ones  which 
take  their  place.  There  is  thus  a  constant  circulation  going  on 
which  tends  to  equalize  the  temperature  of 
the  whole,  by  bringing  the  hot  portions 
into  contact  with  the  colder,  and  thus 
greatly  facilitating  the  conduction  of  heat 
among  the  molecules. 

This  motion  is  made  visible  in  a  glass 
vessel,   by  putting  into    the   water    some 
opaque  powder  of  nearly  the  same  density 
as  water.    Ascending  currents  are  seen  over 
the  part  most  heated,  and  descending  cur- 
rents in  the  parts  farthest  from  the  heat,  as 
represented  in  Fig.  299.     The  ocean  has  per- 
petual currents  caused  in  a  similar  manner. 
The  hottest  portions  flow  away  from  the 
tropical  toward  the  polar  latitudes,  while  ab 
greater  depth  the  cold  waters  of  high  lati- 
tudes flow  back  toward  the  tropics. 
For  a  like  reason,  the  air  is  constantly  in  motion.     The  atmos- 
pheric currents  on  the  earth  have  been  considered  in  Chapter  III. 
of  Pneumatics. 

506.  Determination  of  the  Temperature   of  Water 
at  its  Maximum  Density.— The  apparatus  used  by  Joule  in 
his  research  is  represented  in  outline  in  Fig.  300,  in  which  A  and 
B  are  cylinders  4£  feet  high  and  6  inches  in 
diameter ;  the  open  trough  C  connects  them  at 
top,  and  a  large  tube,  with  stop-cock  D,  con- 
nects them  at  the  bottom.     "When  the  cylinders 
were  filled  so  that  there  was  a  free  flow  through 
the  trough  (?,  any  difference  of  density  in  A 
and  B  would   produce   a   convection   current 
through  D  and   C,  and  the  existence  of  such 
current  in  C  was  made  known  by  the  motion  of 
a  small  glass  bulb,  of  nearly  the  specific  gravity 
of  water,  floating  in  the  trough.     A  very  slight 
difference   of    density   between    the   water  in 
A  and   B,   gave   motion    to   the    bulb   in   C. 
The   cock   D  being  closed,   the    temperatures 
of  A  and  B  were  adjusted  so  that  one  should  be  above  and  the 
other  below  that  of   maximum  density.      Having  recorded  the 
temperatures,  D  was  opened,  and  any  difference  of  density  would 
be  shown  by  a  motion  of  the  bulb  towards  the  denser  column. 
By  carefully  adjusting  the  temperatures,  so  that  upon  opening  D 


RADIATION    OF    HEAT.  325 

no  motion  in  the  trough  C  should  result,  a  pair  of  temperatures 
was  obtained,  corresponding  to  the  same  density.  From  a  series 
of  such  pairs,  the  differences  of  which  were  made  successively 
smaller,  Joule  fixed  the  temperature  of  maximum  density  at 
39.1°  F.  or  3.94°  C.,  very  nearly. 

507.  Radiation  of  Heat. — Radiation  of  heat  is  the  com- 
munication of  the  vibrations  of  the  heated  body  to  the  ether  sur- 
rounding it,  by  which  the  waves  of  heat  are  transmitted  in 
the  manner  already  explained  in  the  article  Light.  Heat  rays 
differ  from  rays  of  light  only  in  wave  length,  and  are  capable  of 
reflection,  refraction,  interference,  and  polarization.  A  body  not 
hot  enough  to  send  forth  rays  affecting  the  optic  nerve,  still  sends 
out  heat  rays,  nor  can  any  body  be  so  cold  as  not  to  radiate  heat 
at  all. 

The  intensity  of  heat  radiated  from  a  given  source,  is  governed 
by  the  three  following  laws  : 

1.  The  intensity  of  radiated  heat  varies  as  the  temperature  of 
the  source. 

2.  It  varies  inversely  as  the  square  of  the  distance. 

3.  It  grows  less,  while  the  inclination  of  the  rays  to  the  surface 
of  the  radiant  grows  less. 

The  truth  of  these  laws  is  ascertained  by  a  series  of  careful 
experiments.  But  the  second  may  be  proved  mathematically 
from  the  fact  of  propagation  in  straight  lines,  as  in  sound  and 
light.  For  the  heat,  as  it  advances  in  every  direction  from  the 
radiant,  is  spread  over  spherical  surfaces  which  increase  as  the 
squares  of  the  distances  ;  therefore  the  intensities  must  grow  less 
in  the  same  ratio ;  that  is,  the  intensities  vary  inversely  as  the 
squares  of  the  distances. 

The  radiating  power  of  a  given  body  depends  on  the  condition 
of  its  surface. 

If  a  cubical  vessel  filled  with  hot  water  have  one  of  its  vertical 
sides  coated  with  lamp  black,  another  with  mica,  a  third  with 
tarnished  lead,  and  the  fourth  with  polished  silver,  and  the 
heat  radiated  from  these  several  sides  be  concentrated  upon  a 
thermometer  bulb,  the  ratio  of  radiation  will  be  found  nearly  as 
follows  : 

Lamp  black 100    I    Tarnished  lead 45 

Mica 80    I    Polished  silver 12 

Polished  metals  generally  radiate  feebly  ;  and  this  explains  the 
familiar  fact  that  hot  liquids  retain  their  temperature  much  better 
in  bright  metallic  vessels  than  in  dark  or  tarnished  ones. 

When  the  temperature  of  a  body  is  gradually  raised,  not  only 


326  HEAT. 

are  new  kinds  of  radiations  produced,  whose  wave  lengths  are 
smaller  than  those  already  emitted,  but  the  intensity  of  existing 
radiations  also  increases.  A  white-hot  body  emits  more  recj  rays 
than  a  red-hot  body,  and  more  non-luminous  rays  than  a  non- 
luminous  body. 

508.  Equalization  of  Temperature. — Radiation  is  going 
on  continually  from  all  bodies,  more  rapidly  in  general  from  those 
most  heated  ;  and  therefore  there  is  a  constant  tendency  toward 
an  equal  temperature  in  all  bodies.     A  system  of  exchange  goes 
on,  by  which  the  hotter  bodies  grow  cool,  and  the  colder  ones 
grow  warm,  till  the  temperature  of  all  is  the  same.     But  this 
equality  does  not  check  the  radiation ;  it  still  goes  forward,  each 
body  imparting  to  every  other  as  much  heat  as  it  receives  from  it, 
the  radiations  emitted  and  absorbed  by  either  body  being  equal 
not  only  in  total  heating  effect,  but  being  the  same  in  the  inten- 
sity, wave  length,  and  plane  of  polarization  of  every  component 
part  of  either  radiation. 

509.  Reflection  of  Heat.— When  rays  of  heat  meet  the 
surface  of  a  body,  some  of  them  are  reflected,  passing  off  at  the 
same  angle  with  the  perpendicular  on  the  opposite  side.     But 
others  pass  into  the  body,  and  are  said  to  be  absorbed  by  it.     It  is 
true  of  waves  of  heat  as  of  all  other  kinds  of  vibration,  that  when 
they  meet  a  new  surface  and  are  reflected,  the  angle  of  incidence 
equals  the  angle  of  reflection,  and  that  their  intensity  after  reflec- 
tion is  weakened. 

If  a  person,  when  near  a  fire,  holds  a  sheet  of  bright  tin  so  as 
to  see  the  light  of  the  fire  reflected  by  it,  he  will  plainly  perceive 
that  heat  is  reflected  also.  And  if  any  sound  is  produced  by  the 
fire,  as  the  crackling  of  combustion,  or  the  hissing  of  steam  from 
wood,  the  reflection  of  the  sound  is  likewise  heard.  This  simple 
experiment  proves  that  waves  of  sound,  of  heat,  and  of  light,  all 
follow  the  same  law  of  reflection. 

510.  Heat  Concentrated  by  Reflection. — Let  two  pol- 
ished reflectors,  M  and  N  (Fig.  301),  having  the  form  of  concave 
paraboloids,  be  placed  ten  or  fifteen  feet  apart,  with  their  axes  in 
the  same  straight  line,  and  let  a  red-hot  iron  ball  be  in  the  focus 
A  of  one,  and  an  inflammable  substance,  as  phosphorus,  in  the 
focus  B  of  the  other ;  then  the  latter  will  be  set  on  fire  by  the 
heat  of  the  ball.     The  rays  diverging  from  A  to  M  are  reflected 
in  parallel  lines  to  N,  and  then  converged  to  B. 

If,  instead  of  phosphorus,  the  bulb  of  a  thermometer  is  put  in 
the  focus  B,  a  high  temperature  is  of  course  indicated  on  the  scale. 
Now  remove  the  hot  ball  from  A,  and  put  in  its  place  a  lump  of 


ABSORPTION     OF    HEAT.  327 

ice ;  then  the  thermometer  at  B  sinks  far  below  the  temperature 
of  the  room.  This  last  experiment  does  not  prove  that  cold  is 
reflected  as  well  as  heat,  but  confirms  what  was  stated  (Art.  508), 

FIG.  301. 


JNT 


that  all  objects  radiate  to  one  another  till  their  temperatures  are 
equalized.  The  ice  radiates  only  a  little  heat,  which  is  reflected 
to  the  thermometer,  but  the  latter  radiates  much  more,  which  is 
reflected  to  the  ice,  so  that  the  temperature  of  the  thermometer 
rapidly  sinks. 

511.  Absorption  of  Heat. — So  much  of  the  radiant  heat 
as  falls  on  a  body  and  is  not  reflected,  is  absorbed.     The  absorbing 
power  in  a  body  is  found  to  be  in  general  equal  to  its  radiating 
power.     It  is  very  noticeable  that  bodies  equally  exposed  to  the 
radiant  heat  of  the  sun  or  a  fire,  become  very  unequally  heated. 
A  white  cloth  on  the  snow,  under  the  sunshine,  remains  at  the 
surface ;  a  black  cloth  sinks,  because  it  absorbs  heat,  and  melts 
the  snow  beneath  it.     Polished  brass  before  a  fire  remains  cold  ; 
dark,  unpolished  iron,  is  soon  hot. 

Lamp  black  reflects  little  of  the  radiation  which  falls  on  it ; 
nearly  the  whole  is  absorbed. 

Polished  silver  reflects  the  greater  part  of  the  radiations  falling 
upon  it,  absorbs  only  about  2J  per  cent.,  and  transmits  none. 

Eock  salt  reflects  less  than  8  per  cent,  of  the  radiation  it 
receives,  absorbs  almost  none,  and  transmits  92  per  cent. 

512.  Diathermancy. — Substances  which  transmit  heat  rays, 
without  themselves  becoming  hot,  are  called  Diathermanous ; 


328  HEAT. 

those  which  are  heated  by  the  transmission  are  said  to  be  AtJier- 
manous. 

Radiant  heat  passes  freely  through  the  atmosphere  as  well  as 
through  vacant  space.  The  air  is  therefore  said  to  be  diathermal ; 
it  is  also  transparent,  since  it  permits  light  to  pass  freely  through 
it.  But  there  are  substances  which  allow  the  free  transmission  of 
the  waves  of  light,  but  not  those  of  heat ;  and  there  are  others 
through  which  waves  of  heat  can  freely  pass,  but  not  those  of 
light. 

Water  and  glass,  which  are  almost  perfectly  transparent  to 
the  faintest  light,  will  not  transmit  the  vibrations  of  heat  unless 
they  are  very  intense.  If  an  open  lamp-flame  shines  upon  a  thin 
film  of  ice,  while  nearly  the  whole  of  the  light  is  transmitted,  only 
6  per  cent,  of  the  heat  can  pass  through. 

A  plate  of  rock  salt,  one-tenth  of  an  inch  thick,  will,  as  shown 
in  the  last  paragraph,  transmit  92  per  cent,  of  the  heat  of  a 
lamp;  and  if  it  be  coated  with  lampblack  so  thick  as  to  stop 
light  completely,  the  heat  is  still  transmitted  with  almost  no 
diminution. 

Prisms  and  lenses  of  rock  salt  have  been  used  in  illustrating 
refraction  of  heat,  just  as  glass  prisms  and  lenses  are  used  in  the 
case  of  luminous  rays, 


CHAPTEE    III. 

SPECIFIC  HEAT.— CHANGES  OF  CONDITION.— LATENT  HEAT. 

513.  Specific  Heat. — The  heat  which  is  absorbed  by  a  body 
is  not  wholly  employed  in  raising  its  temperature.  While  a  part 
of  the  thermal  force  which  is  communicated,  throws  the  atoms 
into  vibration,  that  is,  heats  the  body,  another  part  performs 
interior  work  of  some  other  kind,  such  as  urging  the  atoms 
asunder,  or  forcing  them  into  new  arrangements.  This  latter 
portion  is  lost  to  our  sense  and  to  the  thermometer,  until  the 
body  is  again  cooled,  when  it  reappears.  The  relative  quantity  of 
the  force  thus  hidden  from  view  is  different  in  different  sub- 
stances. Hence  the  phrase  specific  heat,  is  used  to  express  the 
amount  of  heat  required  to  raise  a  given  weight  of  a  substance 
one  degree  of  temperature  as  compared  with  the  amount  required 
to  raise  an  equal  weight  of  water  one  degree.  The  specific  heats 
of  nearly  all  solids  and  liquids  increase  with  the  temperature. 


FINDING    SPECIFIC    HEAT. 


329 


The  specific  heat  of  a  gas  is  nearly  constant  at  all  temperatures 
and  under  all  pressures. 

The  mean  specific  heats  of  a  few  substances,  between  0°  and 
100°  C.  are  given  below  to  show  how  greatly  they  differ : 


Water 1.0000 

Ice... 5040 

Sulphur 2026 

Iron 1138 

Copper 0951 


Silver 0570 

Tin 0562 

Mercury 0333 

Gold 0324 

Lead 0314 


The  specific  heat  of  water  is  greater  than  that  of  every  other 
substance  known,  and  therefore  it  is  made  the  standard  of  com- 
parison. 

The  great  specific  heat  of  water  moderates  the  changes  of 
temperature  upon  islands  and  upon  the  sea-coast.  A  pound  of 
water,  losing  one  degree  of  temperature,  would  raise  about  4.2 
Ibs.  of  air  one  degree,  the  specific  heat  of  air  being  0.2379.  But 
water  is  770  times  as  heavy  as  air ;  hence  the  4.2  Ibs.  of  air  is  3234 
times  the  volume  of  one  pound  of  water. 

The  TJiermal  unit  is  variously  given  as  the  amount  of  heat 
required  to  raise  1  Ib.  of  water  1°  F.,  or  1  Ib.  of  water  1°  C.,  or  1 
kilogramme  of  water  1°  C. 

The  thermal  units  which  we  shall  use  are  the  amounts  of  heat 
necessary  to  raise  the  temperature  of  a  pound  of  water  1°  C.  or 
1°  P.,  as  may  be  indicated  in  the  text. 

514.  Method  of  Finding  Specific  Heat. — The  following 
is  one  of  several  methods  of  finding  the  specific  heat  of  a  sub- 
stance; it  is  called  the  method  of  mixtures. 

Let  a  pound  of  mercury  at  100°  0.  be  poured  into  a  pound  of 
water  at  0°  C.,  and  suppose  the  temperature  of  the  mixture  to  be 
3.2°  C.  Let  x  equal  the  specific  heat  of  mercury.  Now  one 
pound  of  water  has  been  raised  from  0°  C.  to  3.29  C.,  requiring 
for  this  change  3£  thermal  units ;  one  pound  of  mercury  has 
cooled  from  100°  C.  to  3.2°  C.,  thus  giving  out  x  x  96.8  thermal 
units,  x  representing  that  fraction  of  a  thermal  unit  which  would 
raise  one  pound  of  mercury  one  degree ;  as  no  heat  has  been  lost, 
these  two  amounts  must  be  equal,  and  we  have  3.2  =  (96.8)  x9 
from  which  we  find  x  =  .033. 

Suppose  five  pounds  of  iron  at  100°  C.  to  be  put  into  ten  pounds 
of  water  at  10°  C.,  what  will  be  the  temperature  of  the  mixture  ? 
Let  x°  be  the  resulting  temperature.  The  10  Ibs.  of  water  will 
absorb  10  (x  —  10)  units  of  heat ;  the  5  Ibs.  of  iron  will  give  out 
5  (100  —  x)  x  .1138  units  of  heat,  and  these  two  amounts  must 
be  equal;  hence,  10  (x  —  10)  =  5  (100  —  x)  x  .1138,  whence 
x  —  14.8°  C. 


330  HEAT. 

The  specific  heats  of  substances  are  also  found  by  determining* 
the  amounts  of  ice  at  0°  C.,  or  32°  F.,  which  they  will  melt  in 
cooling  from  a  given  temperature  to  that  of  melting  ice. 

The  specific  heat  of  a  substance  in  a  liquid  state  is  generally 
greater  than  in  the  solid  form.  The  specific  heats  of  the  more 
perfect  gases  are  nearly  equal  to  that  of  air,  which  is  0.237. 

515.  Apparent  Conduction  Affected  by  Specific  Heat. 

— The  conducting  power  of  different  substances  cannot  be  correctly 
compared,  without  making  allowance  for  their  specific  heat  (Art. 
501).  For  the  heat  which  is  communicated  to  one  end  of  a  rod, 
will  collect  at  the  other  end  more  slowly,  if  a  great  share  of  it  dis- 
appears on  the  way.  Fo'r  instance,  at  the  same  distance  from  the 
source  of  heat,  wax  is  melted  quicker  on  a  rod  of  bismuth  than  on 
one  of  iron,  though  iron  is  the  best  conductor,  because  the  specific 
heat  of  iron  is  three  times  as  great  as  that  of  bismuth  ;  the  heat 
actually  reaches  the  wax  soonest  through  the  iron,  but  not  enough 
to  melt  it,  because  so  much  is  required  to  raise  the  iron  to  a  given 
temperature. 

516.  Changes  of  Condition.— Among  the  most  important 
effects  produced  by  heat,  are  the  changes  of  condition  from  solid 
to  liquid  and- from  liquid  to  gas,  or  the  reverse,  according  as  the 
temperature  of  a  body  is  raised  or  lowered.     Increase  of  heat 
changes  ice  to  water,  and  water  to  steam,  and  the  diminution  of 
heat  reverses  these  effects.     A  large  part  of  the  simple  substances, 
and  of  compound  ones  not  decomposed  by  heat,  undergo  similar 
changes  at  some  temperature  or  other  ;  and  probably  it  would  be 
found  true  of  all  if  the  requisite  temperature  could  be  reached. 

The  melting  point  (called  also  freezing  point,  or  point  of  conge- 
lation) of  a  substance  is  the  temperature  at  which  it  changes 
from  a  solid  to  a  liquid  or  the  reverse. 

The  boiling  point  is  the  temperature  at  which  it  changes  from 
a  liquid  to  a  gas  or  the  reverse. 

517.  Latent  Heat. — Whenever  a  solid  becomes  a  liquid,  or 
a  liquid  become  a  gas,  a  large  amount  of  heat  disappears,  and  is 
said  to  become  latent.     The  thermal  force  is  expended  in  sunder- 
ing the  atoms,  and  perhaps  in  putting  them  into  new  relations 
and  combinations,  so  that  there  is  not  the  slightest  increase  of 
temperature  after  the  change  begins  till  it  ends.    The  force  is  not 
lost,  but  is  treasured  up  in  the  form  of  potential  energy,  which 
becomes  available  whenever  a  change  is  made  in  the  opposite 
direction.     Using  the  force  of  heat  to  turn  water  into  steam,  is 
like  using  the  strength  of  the  arm  in  coiling  up  a  spring,  or  lifting 
a  weight  from  the  earth.     The  spring  and  the  weight  are  each  in 


LATENT    HEAT. 


331 


a  condition  to  perform  work.  They  have  potential  energy,  which 
can  be  used  at  pleasure. 

It  has  been  already  noticed  that  much  heat  disappears  in 
bodies  of  great  specific  heat,  as  their  temperature  rises.  But  the 
amount  which  becomes  latent,  while  a  change  of  condition  takes 
place,  is  vastly  greater. 

Let  heat  be  supplied  at  a  uniform  rate  to  a  mass  of  water,  n 
pounds,  at  0°  0.,  and  note  the  time  required  to  raise  it  to  100°  C. ; 
continuing  the  same  uniform  supply  of  heat,  it  will  take  5.37 
times  as  long  to  change  the  n  pounds  into  steam  at  100°  0.  In 
raising  the  temperature  from  0°  to  100°  the  number  of  thermal 
units  required  was  100  x  M,  and  in  changing  into  steam  5.37 
times  as  many  thermal  units  were  added,  or  537  x  n  thermal 
units.  The  whole  of  the  537  x  n  thermal  units  have  been  em- 
ployed in  rearranging  the  atoms,  without  producing  any  change 
of  temperature.  The  latent  heat  of  water  is  80 ;  that  is  to  say,  it 
will  require  n  x  80  thermal  units  to  change  n  pounds  of  ice  at 
0°  C.  into  water  also  at  0°  C. 

The  latent  heat  of  steam  is  not  the  same  for  all  temperatures  ; 
the  total  number  of  heat  units,  Q,  required  to  change  n  pounds 
of  water  at  t°  C.  into  steam  at  t°  C.  is  expressed  by  the  formula 

Q=  (606.5  +  0.305  f)  n. 

For  100°  C.,@  =  637  n ;  for  150°  C.,  Q  =  651  n\  for  200°  C., 
Q  =  667.5  n. 

518.  Fusion  or  Melting. — The  change  from  the  solid  to 
the  liquid  state  may  be  either  very  gradual  or  very  abrupt.  As 
the  temperature  rises,  many  substances  become  pasty,  like  wrought 
iron  at  white  heat,  and  for  a  considerable  range  of  temperatures 
such  substances  are  neither  solid  nor  liquid,  and  no  definite  melt- 
ing point  can  be  assigned.  Ice  passes  very  abruptly  from  the 
solid  to  the  liquid  state,  probably  during  a  rise  of  temperature  not 
greater  than  0.1°  C. 

From  the  beginning  of  fusion  till  the  end  of  the  change  of 
condition  there  is  no  rise  of  temperature,  the  heat  which  does 
internal  work  being  termed  latent  heat  of  fusion. 

The  latent  heat  of  fusion  of  ice  has  already  been  given  (Art 
517).  The  melting  points  of  a  few  substances  are  given  below  : 

Ice 0°  C. 

Spermaceti 49°  C. 

White  Wax 65°  C. 

Sulphur 111°  C. 

The  melting  point  of  a  substance  which  expands  on  solidify- 
ing is  lowered  by  great  increase  of  pressure  above  the  ordinary 


Tin 235°  C. 

Lead 325°  C. 

Silver 1000°  C. 

Iron..  ..1500°C. 


332  HEAT. 

pressure  of  the  atmosphere,  while  that  of  a  substance  which  con- 
tracts in  solidifying  is  raised.  The  melting  point  of  wax  was 
raised  from  65°  C.  to  75°  C.  by  a  pressure  of  520  atmospheres, 
while  the  melting  point  of  ice  is  lowered  about  0.0074°  C.  for 
every  additional  pressure  of  one  atmosphere. 

Alloys  are  generally  more  fusible  than  the  metals  of  which 
they  are  composed. 

519.  Vaporization. — The  change  from  the  liquid  to  the 
gaseous  state  is  termed  vaporization.     This  change  is  sometimes 
effected  quietly  without  the  formation  of  bubbles,  then  termed 
evaporation,  and  sometimes  in  a  violent  manner  with  the  forma- 
tion of  bubbles,  to  which  action  the  term  ebullition  is  applied. 

Vaporization  is  more  rapid  as  the  pressure  upon  the  surface 
of  the  liquid  is  diminished. 

The  boiling  point  of  water  at  one  atmosphere,  at  the  level  of 
the  ocean,  is  100°  C.;  but  upon  the  tops  of  high  mountains  the 
boiling  point  is  90°  and  85°  C.,  and  in  the  air  pump  vacuum  it  is 
as  low  as  23°  C. 

The  effect  of  diminished  pressure  to  lower  the  boiling  point  is 
well  shown  by  the  following  familiar  experiment :  In  a  thin  glass 
flask,  boil  a  little  water,  and  after  removing  it  from  the  fire,  cork 
and  invert  the  flask.  The  steam  which  is  formed  will  soon  press 
so  strongly  upon  the  water  as  to  stop  the  boiling.  When  this 
happens,  pour  a  little  cold  water  upon  the  flask ;  the  water  within 
will  immediately  commence  boiling  violently,  because  the  vapor 
is  condensed  and  the  pressure  removed.  This  effect  may  be 
reproduced  several  times  before  the  water  in  the  flask  is  too  cool 
to  boil  in  a  vacuum. 

520.  Other  Causes  Affecting  the  Boiling  Point.— The 

boiling  point  is  raised  by  substances  in  solution,  provided  they  are 
less  volatile  than  the  liquid  in  which  they  are  dissolved. 

Water  saturated  with  common  salt  boils  at  109°  C.,  and  when 
chloride  of  calcium  replaces  the  salt  the  boiling  point  is  raised  to 
179°  C.  Substances  held  in  suspension,  but  not  dissolved,  have 
no  effect  upon  the  boiling  point. 

Water  from  which  the  dissolved  air  has  been  removed  by 
previous  ebullition,  has  been  raised  to  112°  C.  before  boiling,  the 
elastic  air  seeming  to  act  as  a  spring  to  aid  ebullition. 

Water  boils  at  a  higher  temperature  in  glass  vessels  than  in 
metallic  ones,  rising  as  high  as  105°  C.  before  ebullition  begins. 
If  metal  clippings  or  filings,  or  any  angular  fragments  whatever 
which  may  serve  as  a  nucleus,  be  dropped  into  the  flask,  the  boiling 
point  is  brought  down  to  100°  C.,  and  the  violent  bumping  which 
accompanies  ebullition  at  the  higher  temperatures  is  prevented. 


EVAPORATION.  333 

521.  Spheroidal  Condition. — When  a  little  water  is  placed 
in  a  red-hot  metallic  cup,  instead  of  boiling  violently,  and  disap- 
pearing in  a  moment,  as  might  be  expected,  it  rolls  about  quietly 
in  the  shape  of  an  oblate  spheroid,  and  wastes  very  slowly.  So 
drops  of  water,  falling  on  the  horizontal  surface  of  a  very  hot 
stove,  are  not  thrown  off  in  steam  and  spray  with  a  loud  hissing 
sound,  as  they  are  when  the  stove  is  only  moderately  heated,  but 
roll  over  the  surface  in  balls,  slowly  diminishing  in  size  till  they 
disappear. 

In  such  cases,  the  water  is  said  to  be  in  the  spheroidal  state. 
Not  being  in  contact  with  the  metal,  it  assumes  the  shape  of  an 
oblate  spheroid,  in  obedience  to  its  own  molecular  attractions  and 
the  force  of  gravity,  as  small  masses  of  mercury  do  on  a  table. 
The  reason  why  the  water  does  not  touch  the  hot  metal  is,  that 
the  heat  causes  a  coat  of  vapor  to  be  instantly  formed  about  the 
drop,  on  which  it  rests  as  on  an  elastic  cushion  ;  and  as  the  vapor 
is  a  poor  conductor  of  heat,  further  evaporation  proceeds  very 
slowly.  It  is  easily  seen  that  the  spheroid  does  not  touch  the 
metal,  by  so  arranging  the  experiment  that  a  beam  of  light  may 
shine  horizontally  upon  the  drop,  and  cast  its  shadow  completely 
separated  from  that  of  the  hot  plate  below  it,  as  in  Fig.  302. 

FIG.  302. 


If  the  heated  surface  is  cooling,  the  temperature  may  become 
so  low  that  the  drop  at  length  touches  it,  when  in  an  instant 
violent  ebullition  takes  place,  and  the  water  quickly  disappears  in 
vapor. 

622.  Evaporation. — Many  liquids  and  even  solids  pass  into 
the  gaseous  state  by  a  slow  and  almost  insensible  process,  which 
goes  on  at  the  surface.  This  is  called  evaporation  ;  and  it  takes 
place  at  all  temperatures,  but  more  rapidly  as  the  temperature  is 
higher.  Ice  and  snow  waste  away  gradually  at  temperatures  far 
below  0°  C.,  and  the  odor  of  brass,  copper,  and  iron  is  attributed 
to  an  insensible  evaporation  of  these  metals. 

523.  Condensation. — The  change  from  the  condition  of 
vapor  to  the  liquid  state  is  called  condensation.  This  change  of 
state  may  be  caused  by  cooling  and  by  compression.  A  saturated 


334  HEAT. 

vapor  at  any  given  temperature  and  pressure  will  be  partially  con- 
densed by  either  lowering  the  temperature  or  by  increasing  the 
pressure.  Those  gases  which  have  usually  been  called  permanent 
gases,  because,  under  ordinary  conditions  they  are  very  far  re- 
tnoved  from  their  point  of  condensation,  have  been  reduced  to 
the  liquid  state  by  very  low  temperatures  and  great  pressures 
combined. 

Vapors  give  up  their  latent  heat  of  vaporization  during  the 
process  of  condensation  ;  the  latent  heat  of  steam  may  be  deter- 
mined by  passing  a  known  weight  of  steam  at  100°  C.  into  a  given 
quantity  of  water  at  a  known  temperature,  and  taking  the  result- 
ing temperature. 

Suppose  1  Ib.  of  steam  at  100°  C.  to  be  condensed  by  6  Ibs.  of 
water  at  0°  C.,  and  that  the  resulting  temperature  is  91°  C.  The 
6  Ibs.  of  water  raised  from  0°  to  91°  required  6  x  91  —  54G  heat 
units  ;  the  pound  of  steam,  after  condensation  at  100°,  gave  up  9 
of  these  546  units  in  cooling  from  100°  to  91°,  leaving  537  heat 
units  as  the  latent  heat  of  vaporization. 

524.  Solidification.— Substances  which  have  been  melted 
and  which  cool  slowly  while  passing  into  the  solid  state  usually 
assume  a  regular  crystalline  structure.    If  they  expand  on  solidify- 
ing, the  solid  will  float  in  the  liquid,  but  if  they  contract  the 
solid  will  sink. 

A  liquid  may  be  cooled  below  its  normal  temperature  of  solidi- 
fication. A  hot  saturated  solution  of  Glauber's  salt,  cooled  slowly 
and  at  rest,  will  remain  liquid  at  the  ordinary  temperature  of  the 
atmosphere ;  but  upon  being  suddenly  jarred,  or  when  a  crystal 
of  the  salt  is  dropped  into  the  liquid,  the  molecular  equilibrium 
is  destroyed  and  solidification  ensues  at  once.  Water  which  has 
been  boiled,  to  free  it  from  air,  may  be  cooled  to  —10°  C.,  or 
even  lower,  without  freezing;  but  any  vibration  causes  instant 
crystallization.  In  all  such  cases  the  latent  heat  of  fusion  becomes 
sensible,  and  may  be  felt  by  placing  the  hands  upon  the  contain- 
ing vessel. 

The  freezing  point  of  water  containing  salt  in  solution  is 
lower  than  that  of  pure  water.  Sea  water  freezes  at  —2.5°  C.  to 
— 3°  C.  ;  the  ice  is  pure,  containing  none  of  the  salt. 

525.  Freezing    Produced   by   Melting.— Since  a  great 
amount  of  heat  disappears  in  a  substance  as  it  passes  from  the 
solid  to  the  liquid  state,  the  loss  thus  occasioned  may  produce 
freezing  in  a  contiguous  body.     When  salt  and  powdered  ice  are 
mixed,  their  union  causes  liquefaction.     And  if  this  mixture  is 
surrounded  by  bad  conductors,  and  a  tin  vessel  containing  some 


FREEZING     BY    EVAPORATION.  335 

liquid  be  placed  in  the  midst  of  it,  the  latter  is  frozen  by  the 
abstraction  of  heat  from  it,  by  the  melting  of  the  ice  and  salt.  In 
this  way  ice  creams  and  similar  luxuries  are  easily  prepared  in 
hot  as  well  as  in  cold  weather. 

526.  Freezing  by  Evaporation, — In  like  manner,  freezing 
by  evaporation  is  explained.     Put  a  little  water  in  a  shallow  dish 
of  thin  glass,   and  set  it  on  a  slender  wire-support  under  the 
receiver  of  an  air  pump.     Beneath  the  wire-support  place  a  broad 
dish  containing  sulphuric  acid.    When  the  air  is  exhausted,  the 
water  in  a  few  moments  is  found  frozen.    As  the  pressure  of  the 
air  is  taken  off,  evaporation  proceeds  with  increased  rapidity,  and 
the  requisite  heat  for  this  change  of  condition  can  be  taken  only 
from  the  dish  of  water.     But  the  atmosphere  of  vapor  retards  the 
process  by  its  pressure  ;  hence  the  sulphuric  acid  is  placed  in  the 
receiver,  so  as  to  seize  upon  the  vapor  as  fast  as  formed,  and  thus 
render  the  vacuum  more  complete.    The  water  is  frozen  by  giving 
up  its  heat  to  become  latent  in  the  vapor,  so  rapidly  formed  ;  but 
when  this  vapor  becomes  liquid  again  in  combining  with  the  acid, 
the  same  heat  reappears  in  raising  the  temperature  of  the  acid. 

Thin  cakes  of  ice  may  sometimes  be  procured,  even  in  the 
hottest  climates,  by  the  evaporation  of  water  in  broad  sh allow 
pans  under  the  open  sky,  where  radiation  by  night  aids  in  redu- 
cing the  temperature.  The  pans  should  be  so  situated  as  to  receive 
the  least  possible  heat  by  conduction. 

Various  ice-making  machines  have  been  devised  in  which  the 
vaporization  of  some  volatile  liquid,  such  as  ether,  liquid  am- 
monia, liquid  sulphurous  acid,  &c.,  abstracts  sufficient  heat  from 
the  water  to  freeze  it. 

527.  Regelation.— If  two  pieces  of  ice  at  0°  C.,  having 
smooth  surfaces,  be  pressed  together,  they  will  soon  adhere,  and 
will  do  this  in  air,  in  water,  or  in  vacuo.     This  freezing  together 
again  is  called  regelation. 

The  interior  of  a  block  of  melting  ice  is  a  little  colder  than  the 
surface  :  now  when  the  two  surfaces  are  pressed  together,  the  very 
thin  film  of  water  which  covers  them  is  removed  from  the  warmer 
air,  and  is  in  the  same  condition  as  though  transferred  to  the 
interior  of  a  block,  the  lower  temperature  of  which  freezes  it. 


336 


HEAT 


CHAPTER    IV. 

TENSION    OF  VAPOR.— THE    STEAM-ENGINE. -MECHANICAL 
EQUIVALENT    OP  HEAT. 

528.  Dalton's  Laws.— 

1.  Whatever  be  the  temperature  of  a  liquid  which  partly  fills 
a  vessel,  vaporization  will  go  on  till  the  vessel  is  filled  with  vapor, 
of  a  density  determined  solely  by  the  temperature,  after  which 
vaporization  will  cease. 

2.  If  the  space  occupied  by  the  vapor  be  made  larger,  the  tem- 
perature being  the  same,  then  vaporization  will  again  go  on  till 
the  density  is  the  same  as  before.     If  the  space  be  made  smaller, 
the  temperature  remaining  constant,  a  part  of  the  vapor  returns 
to  the  liquid  state,  and  the  remaining  vapor  will  have  the  same 
density  as  before. 

3.  If,  besides  the  liquid  and  its  vapor,  the  vessel  contains  any 
gas,  not  capable  of  chemical  action  on  the  liquid,  then  exactly  the 
same  amount  of  vapor,  of  the  same  density  as  before,  will  be 
formed ;  but  the  time  required  to  reach  the  maximum  density 

will  be  greater  because  of  the  mechanical  obstruc- 
tion to  a  rapid  diffusion,  which  the  gas  offers. 

A  vapor  at  the  maximum  density  and  pressure 
for  the  given  temperature  is  called  a  saturated 
vapor. 

529.  Experimental  Illustration. — Fill  a 
barometer  tube  A  B  (Fig.  303)  full  of  mercury ; 
close  the  open  end  with  the  finger  and  invert  into 
the  cup  H  of^he  deep  mercury  cistern  H  K. 
With  a  pipette,  the  tube  of  which  is  bent  upwards 
at  the  end,  transfer  enough  ether  to  the  barometer 
tube  to  leave  a  thin  film  of  liquid  c  d,  after  the 
space  A  d  is  filled  with  saturated  vapor.  Measure 
the  height  c  H  of  the  mercury  column.  If  the 
tube  A  B  be  raised,  tending  to  increase  the  space 
A  d  above  the  liquid,  more  vapor  will  form  and 
c  H  will  remain  unaltered ;  if  the  tube  A  B  be 
depressed,  tending  to  diminish  the  space  A  d, 
vapor  will  condense  to  liquid  again,  and  c  H  will 
still  be  unaltered. 

To  show  the  effect  of  change  of  temperature  use  a  barometer 


Fro 


TENSIONS    OF     VAPORS. 


337 


FIG.  804. 


FIG.  305. 
A.  B   C 


tube  bent  at  its  closed  end  as  in  Fig.  304,  so  that  a  portion  of  the 
bend  may  either  be  surrounded  with  cooling  mixtures,  as  at  A, 
or  may  be  warmed  by  a  flame.  Upon  raising  the  temperature  of 
the  contained  vapor  its  ten- 
sion will  increase  and  the 
mercury  column  C  B  will  be 
shortened ;  upon  lowering 
the  temperature  the  tension 
will  decrease,  and  C  B  will 
lengthen  as  the  mercury 
rises. 

530.  Tensions  of  Dif- 
ferent Vapors.  —  Transfer 
to  three  barometer  tubes,  A, 
B,  and  C  (Fig.  305),  water, 
alcohol  and  ether  respectively, 
and    the    mercury    columns 
will  stand  at  different  heights 
«,  b  and  c,  showing  that  the 
tensions  of  the  three  vapors 
are  not  the  same  at  the  same 
temperature. 

531.  Tension  in  Gen- 
erator and  Condenser.— 

Let  two  vessels,  A  and  B  (Fig.  306),  be  connected  by  a  pipe 
furnished  with  a  stop-cock  C.  Let  the  tubes  a  and  b  be  connected 
with  separate  manometers  to  indicate  the  tensions  of  the  vapor  in 
A  and  B  when  C  is  closed.  Having  partly  filled  A  with  water, 

cause  it  to  boil  until  all  air 
has  been  driven  from  both 
flasks  through  C  and  the 
loosened  stopper  of  B ;  now 
close  B  and  remove  the  lamp. 
The  two  manometers  will 
indicate  the  same  tension  in 
both  flasks. 

Now  close  C  and  surround 
B  with  cold  water ;  part  of 
the  vapor  in  it  will  be  con- 
densed, the  remainder  hav- 
ing a  greatly  reduced  tension  as  shown  by  the  fall  of  the  mercury 
column  in  its  manometer.     Apply  heat  to  A,  thus  forming  new 
vapor  of  higher  tension  than  before,  as  will  be  shown  by  its 
manometer  reading. 
22 


FIG.  306. 


338 


HEAT. 


If  now  C  be  opened  the  manometer  connected  with  A  will  fall 
to  the  same  reading  as  that  of  B,  and  the  two  will  indicate  this 
same  reading  just  as  long  as  the  temperature  of  B  is  kept  con- 
stant and  below  that  of  A. 

The  liquid  in  A  merely  distills  over  to  B,  at  the  tension  of  the 
vapor  in  the  colder  vessel. 

532.  Thermal   Force  in    Steam.— It  has  been    already 
noticed  that  while  water  is  heated,  and  especially  while  it  is  con- 
verted into  steam  by  boiling,  the  heat  apparently  lost  is  so  much 
force  treasured  up  ready  for  use,  as  truly  as  when  strength  is 
expended  in  lifting  great  weights,  which  by  their  descent  can  do 
the  work  desired.     In  modern  engineering,  the  force  of  steam  is 
employed  more  extensively,  and  for  more  varied  purposes,  than 
any  other.     Every  steam-engine  is  a  machine  for  transforming  the 
internal  motion  of  heated  steam  into  some  of  the  visible  forms  of 
motion. 

533.  Tension  of  Steam. — When  steam  is  formed  by  boiling 
water  in  the  open  air,  its  tension  is  equal  to  that  of  the  air,  and 
therefore  ordinarily  about  fifteen  pounds  to  the  square  inch.     But 
when  it  is  formed  in  a  tight  vessel,  so  that  it  cannot  expand,  as 
the  temperature  of  the  water  is  raised  the  tension  is  increased  in 
a  much  greater  ratio  ;  because  the  same  steam  has  greater  tension 
at  a  higher  temperature,  and  besides  this,  new  steam  is  continually 
added. 

The  following  table  gives  the  temperature  corresponding  to 
various  atmospheres  of  tension  : 

Atmosphere.  Temperature. 

)14 F.  384        C.  195.5 

\15 390  198.8 

19 411  210.1 

20..  ..  415  213.0 


It  is  seen  by  the  above  table  that  37°  F.  or  20.6°  0.  are  required 
to  add  the  second  atmosphere  of  tension,  while  only  4°  F.  or 
2.6°  C.  are  required  to  add  the  twentieth  atmosphere. 

534.  Relation  of  Temperature,  Pressure  and  Volume. 

— The  specific  heat  of  water  is  not  constant,  but  varies  with 
the  temperature  (Art.  513),  and  hence  we  cannot  assume  that 
180°  F.  heat  units  are  required  to  raise  1  Ib.  of  water  from  32°  F. 
to  212°  F.,  nor  that  250  heat  units  will  suffice  to  raise  1  Ib.  of 
water  from  32°  F.  to  282°  F. ;  in  the  first  case  180.9  heat  units 
are  required,  and  in  the  second  252.2  units. 


Atmosphere. 
I  1 

Tempera 
F   212 

ture. 
C.  100 
120.6 
144.0 
152.2 
175.8 
180.3 

\l.:::::: 

249 

(4 

291 

5 

.  306 

(  9 

348 

io.:::::: 

..  356 

STEAM    ENGINES. 


339 


The  latent  heat  of  vaporization  is  also  variable,  being  965.7 
heat  units  at  212°  F.,  and  939.4  units  at  249°  F. 

'  A  formula  for  the  latent  heat  of  vaporization  of  water,  derived 
from  Regnault's  experiments,  is  R  =  1091.7  —  0.695  (t  —  32),  in 
which  R  =  number  of  heat  units  (F. )  required  to  convert  1  Ib.  of 
water  at  t°  F.  into  steam  at  that  temperature.  The  total  heat 
units  (F.)  required  to  raise  1  Ib.  of  water  from  32°  F.  to  t°  F.,  and 
evaporate  it  at  that  temperature  is 

.       L  =  1091.7  +  0.305  (t  —  32). 

The  volume  of  steam  at  t°  F.  as  compared  with  the  volume  of 
water  at  39.2  F.  which  gave  it  is  found,  very  nearly,  from 
V—  1+Q-Q°2Q4(^  —  32) 

0.000055  p 

p  being  the  pressure,  in  pounds  per  square  inch,  corresponding  to 
the  temperature  t°  F. 

The  following  table  illustrates  the  various  relations  which  have 
just  been  discussed  : 


§ 

!i 

1 
|j 
31 

a 
1*. 

ill 

IIS!  I 

M1 

iftft 

it 

?; 

p 

sll 

BO  °  «.2 

•*o     g5-|^ 

I 

0    CD 

& 

o^'&g 

"or--    £ 

B  v  f>^^  ^ 

i 

1 

y 

o^ 

o^ 

oSB^'Sfe 

i 

14.7 

212.0 

180.90 

965.7 

1691 

2 

29.4 

249.1 

218.55 

939.4 

892 

3 

44.1 

273.1 

242.98 

922.2 

615 

4 

58.8 

291.2 

261.56 

909.2 

473 

5 

73.5 

306.0 

276.73 

898.5 

386 

6 

88.2 

318.6 

289.69 

889.4 

327 

7 

102.9 

329.6 

301.04 

881.4 

284 

8 

117.6 

339.4 

311.20 

874.3 

252 

535.  The  Steam-Engines  of  Savery  and  Newcomen. 

— The  only  steam-engines  that  were  at  all  successful  before  the 
great  improvements  made  by  Watt,  were  the  engine  of  Savery  and 
that  of  Newcomen.  ISio  other  purpose  was  proposed  by  either  than 
that  of  removing  water  from  mines. 

In  the  engine  of  Savery,  steam  was  made  to  raise  water  by  act- 
ing on  it  directly,  and  not  through  the  intervention  of  machinery. 

It  consisted  of  a  boiler  B  (Fig.  307) ;  a  cylinder  A,  with  a 
valve  at  c  opening  inward,  and  one  at  d  opening  outward;  a  pipe 
e  to  discharge  cold  water  upon  the  cylinder,  and  a  steam  pipe/, 
from  the  boiler  to  the  cylinder. 

First  the  steam-cock  at /is  opened  and  steam  fills  the  cylinder 
A,  driving  the  air  out  through  the  valve  d.  Next /is  closed  and 


340 


HEAT. 


the  cock  e  is  opened,  allowing  cold  water  to  flow  over  the  cylinder 
from  the  delivery  pipe  0,  thus  condensing  the  steam  in  A,  and 
creating  a  vacuum,  into  which  the  atmosphere  forces  water  from 
the  supply  P,  through  the  valve  c.  Now  e  is  closed  and /opened 

PIG.  307.  FIG.  308. 

O 


again,  and  steam  enters  the  cylinder  A  and  drives  the  water  out 
through  d.  When  A  is  full  of  steam  the  operation  is  repeated  as 
before. 

Newcomen  used  steam  to  work  a  common  pump.  The  weighted 
pump  rod  R  (Fig.  308)  was  attached  to  one  end  of  a  working 
beam  B,  while  at  the  other  end  of  B  was  hung  the  piston  P, 
working  steam  tight  in  the  cylinder  C.  Steam  at  atmospheric 
pressure  from  the  boiler  G  enters  C  through  the  cock  a,  and  P 
being  pressed  upon  equally  on  both  sides  is  drawn  to  the  top  of  C 
by  the  weight  of  the  pump  rod  R.  Now  a  is  closed  and  e  is 
opened,  permitting  cold  water  from  a  tank  T  to  flow  into  C, 
which  condenses  the  steam,  creating  a  vacuum,  and  allows  the 
piston  to  descend  under  atmospheric  pressure.  When  P  has 
reached  the  bottom  of  C,  e  having  been  closed,  a  and  d  are  opened 
and  steam  enters  the  cylinder,  while  the  injection  water  from  T 
flows  out  through  d,  and  the  piston  P  rises  as  at  first.  On  closing 
a  and  d  and  opening  e  the  stroke  is  repeated. 

As  the  water  was  raised  by  the  direct  pressure  of  the  atmos- 
phere, this  invention  of  Newcomen  was  called  the  atmospheric 
engine. 

In  these  diagrams  of  Savery's  and  Newcomen's  engines,  all 
details  of  valves  or  other  working  parts  have  been  omitted,  that 
the  principle  alone  might  claim  attention. 


DOUBLE-ACTING    ENGINE. 


341 


In  neither  of  these  methods  was  steam  used  economically  as  a 
power.  The  movements  in  both  cases  were  sluggish,  and  a  large 
part  of  the  force  was  wasted,  because  the  steam  was  compelled 
to  act  upon  a  cold  surface,  which  condensed  it  before  its  work 
was  done. 

536.  The  Steam-Engine  of  Watt.— Steam  did  not  give 
promise  of  being  essentially  useful  as  a  power  till  Watt,  in  the 
year  1760,  made  a  change  in  the  atmospheric  engine,  which  pre- 
vented the  great  waste  of  force.     Newcomen  introduced  the  cold 
water  which  was  to  condense  the  steam  into  the  steam  cylinder 
itself ;  and  the  cylinder  must  be  cooled  to  a  temperature  below 
100°  F.,  else  there  would  be  steam  of  low  tension  to  retard  the 
descent  of  the  piston.     But  when  the  piston  was  to  be  raised, 
the  cylinder  must  be  heated  again  to  212°  F.,  in  order  that  the 
admitted  steam  might  balance  the  pressure  of  the  air. 

In  the  engine  of  Watt,  the  steam  is  condensed  in  a  separate 
vessel  called  the  condenser.  The  steam  cylinder  is  thus  kept  at 
the  uniform  temperature  of  the  steam.  In  the  first  form  which 
he  gave  to  his  engine,  he  so  far  copied  the  atmospheric  engine  as 
to  allow  the  piston,  after  being  pressed  down  by  steam,  to  be  raised 
again  by  the  load  on  the  opposite  end  of  the  great  beam,  while  the 
steam  circulates  freely  below  and  above  the  piston.  This  was 
called  the  single-acting  engine, 
and  might  be  successfully  used 
for  the  only  purpose  to  which  any 
steam-engine  was  as  yet  applied, 
namely,  pumping  water  from 
mines.  But  he  almost  imme- 
diately introduced  the  change  by 
which  the  whole  force  of  the 
steam  was  brought  to  act  on  the 
upper  and  the  under  side  of  the 
piston.  It  thus  became  double- 
acting,  and  the  steam  force  was 
no  longer  intermittent. 

537.  The    Double-acting 

Engine.— Let   8  (Fig.  309)  be 

the  steam  cylinder,  P  the  piston, 

A  the  piston   rod,  passing  with 

steam-tight  joint  through  the  top 

of  the  cylinder,  C  the  condenser, 

kept  cold  by  the  water  of  the 

cistern  G,   B  the  steam  pipe  from  the  boiler,  K  the  eduction 

pipe,  which  opens  into  the  valve  chest  at  0,  D  D  tlic  D-valve, 


FIG.  309. 


342 


HEAT. 


E  F  the  openings  from  the  valve-chest  into  the  cylinder.  As  the 
D-valve  is  situated  in  the  figure,  the  steam  can  pass  through  B 
and  E  into  the  cylinder  below  the  piston,  while  the  steam  above 
the  piston  can  escape  by  F  through  0  and  K  to  the  condenser, 
where  it  is  condensed  as  fast  as  it  enters ;  so  that  in  an  instant 
the  space  above  the  piston  is  a  vacuum,  while  the  whole  force  of 
the  steam  is  exerted  on  the  under  side.  The  piston  is  therefore 
driven  upward  without  any  force  to  oppose  it.  But  before  it 
reaches  the  top,  the  D-valve,  moved  by  the  machinery,  begins  to 
descend,  and  shut  off  the  steam  from  E  and  admit  it  to  F,  and, 
on  the  other  hand,  to  shut  F  from  the  eduction  pipe  0,  and  open 
E  to  the  same.  The  steam  will  then  press  on  the  top  of  the 
piston,  and  there  will  be  a  vacuum  below  it,  so  that  the  piston 
descends  with  the  whole  force  of  the  steam,  and  without  resist- 
ance. To  render  the  condensation  more  sudden,  a  little  cold 
water  is  thrown  into  the  condenser  at  each  stroke  through  the 
pipe  H. 

538.  Condensing  Engine. — The  principle  of  the  condens- 
ing engine  is  illustrated  by  the  figure  and  description  of  the 
preceding  article.  But  the  condensing  apparatus  of  this  kind  of 
engine  requires  many  other  parts,  most  of  which  are  presented  in 
Fig.  310.  C  is  the  steam  cylinder ;  R  the  rod  connecting  its 

FIG.  310 


piston  with  the  end  of  the  working  beam,  not  represented  ;  A  the 
steam-pipe  and  throttle-valve;  B  B  the  D-valve;  D  D  the  educ- 
tion pipe,  leading  from  the  valve-chest  to  the  condenser  E ;  G  (r 


ESTIMATION    OF    STEAM    POWER.  343 

the  cold  water  surrounding  the  condenser ;  F  the  air-pump,  which 
keeps  the  condenser  clear  of  air,  steam,  and  water  of  condensa- 
tion ;  /  the  hot  well,  in  which  the  water  of  condensation  is  de- 
posited by  the  air-pump ;  K  the  hot-water  pump,  which  forces 
the  water  in  the  hot  well  through  L  to  the  boiler ;  H  the  cold- 
water  pump,  by  which  water  is  brought  to  the  cistern  G  G ;  the 
rods  of  all  the  pumps,  F,  K,  and  H,  are  moved  by  the  working 
beam ;  P  the  fly-wheel ;  M  the  crank  of  the  same,  N  the  connect- 
ing-rod, by  which  the  working  beam  conveys  motion  to  the  fly- 
wheel ;  Q  the  excentric  rod,  by  which  the  D-valve  is  moved  ;  0  0 
the  governor,  which  regulates  the  throttle- valve  in  the  steam- 
pipe  A. 

There  is  much  economy  of  fuel  and  saving  of  wear  in  the 
machinery,  arising  from  the  proper  adjustment  of  the  valves.  If 
the  steam  enters  the  cylinder  during  the  whole  length  of  a  stroke 
of  the  piston,  its  motion  is  accelerated ;  and  is  therefore  swiftest 
at  the  instant  before  being  stopped  ;  thus  the  machinery  receives 
a  violent  shock.  If  the  valve  is  adjusted  to  cut  off  the  steam 
when  the  piston  has  made  one-third  or  one-half  of  its  stroke,  the 
diminishing  tension  may  exert  about  force  enough,  during  the 
remaining  part,  to  keep  up  a  uniform  motion.  The  cut-off,  how- 
ever, should  be  regulated  in  each  engine,  according  to  friction 
and  other  obstructions. 

539.  Non-condensing    Engine. — For    many    purposes, 
especially  those  of  locomotion,  it  is  advantageous  to  dispense  with 
the  large  weight  and  bulk  of  machinery  necessary  for  condensa- 
tion, and  do  the  work  with  steam  of  a  higher  tension.     If  (Fig. 
310)  the  condenser,  cistern,  and  all  the  pumps  are  removed,  then 
the  steam  is  discharged  from  E  and  F  at  each  stroke  into  the  air. 
Therefore  the  steam  in  that  part  of  the  cylinder  which  is  open  to 
the  air,  will  have  a  tension  of  15  Ibs.  per  inch  ;  and,  consequently, 
the  steam  on  the  opposite  side  of  the  piston  must  have  a  ten- 
sion 15  Ibs.  per  inch   greater   than   before,  in   order  to  do  the 
same  work. 

Steam  of  a  pressure  not  greater  than  45  Ibs.  per  inch  (above  the 
atmosphere)  is  called  low  pressure  steam,  or  low  steam  ;  high  steam 
is  at  a  pressure  above  this,  and  not  uncommonly  runs  higher  than 
200  Ibs.  per  inch  by  the  gauge. 

540.  Estimation  of  Steam  Power. — It  is  customary  to 
express  the  power  of  a  steam-engine  by  comparing  the  work  done 
by  it  with  that  which  horses  can  do.     In  making  this  comparison 
Watt  took  as  a  measure  of  one  horse-power  the  ability  to  raise 
2,000,000  Ibs.   through  the  height  of  one  foot  in  an  hour;  or 


344 


HEAT. 


2,000,000  foot-pounds  per  hour.  The  modern  expression  for  one 
horse-power  is  33,000  ft.  Ibs.  per  minute,  or  1,980,000  ft.  Ibs. 
per  hour.  The  horse-power  of  an  engine  varies  with  the  pressure 
of  steam  used,  and  with  the  speed  at  which  it  is  run  ;  hence  it 
is  absurd  to  assign  a  special  horse-power  to  an  engine  unless  the 
conditions  of  the  pressure  and  speed  are  given  also. 

541.  Mechanical  Equivalent  of  Heat. — In  all  cases  in 
which  mechanical  force  produces  heat,  and  again  in  all  those  in 
which  heat  produces  visible  motion,  careful  experiment  proves 
that  heat  and  mechanical  force  may  each  be  made  a  measure  of 
the  other.  Forces  of  any  kind  may  be  compared,  by  observing 
the  weights  which  they  will  lift  through  a  given  distance.  The 
mechanical  equivalent  of  heat  (commonly  called,  from  the  name 
of  an  English  experimenter,  Joule's  equivalent)  is  given  in  the 
following  statement : 

The  force  required  to  heat  one  pound  of  water  one  degree  P.,  is 
equal  to  that  which  would  lift  772  pounds  the  distance  of  one  foot, 
or  is  equal  to  772  foot-pounds. 

The  mode  of  determining  this  value  of  the  mechanical  equiva- 
lent is  the  following : 

A  weight  W  (Fig.  311),  by  means  of  a  cord  passing  over  a 
pully  p  and  around  a  drum  D,  gives  to  the  vertical  axis  A  a  rapid 

rotation.  Attached  to  this  axis 
are  a  number  of  radial  arms,  or 
paddles,  as  shown  in  the  figure ; 
projecting  from  the  sides  of  the 
cylinder  C,  in  which  these  arms 
rotate,  are  fixed  arms,  as  shown,  to 
arrest  any  tendency  to  a  rotary 
motion  of  the  water  in  the  cylin- 
der. 

If  one  pound  of  water  at  60°  F. 
be  put  into  the  cylinder  C,  it  will 
require  the  expenditure  of  772 
foot-pounds  of  energy  on  the  part 
of  the  falling  weight  W  to  raise  its 
temperature  by  agitation  to  61°  F. 

The  force  requisite  to  raise  one  pound  of  water  1°  F.,  is  some- 
times called  the  thermal  unit  (Art.  513),  and  all  forces  may  be 
brought  to  this  as  a  standard  of  comparison.  Thus,  one  horse- 
power (2,000,000  foot-pounds  per  hour)  is  2,590  thermal  units  per 
hour,  or  about  43  per  minute. 

Since  a  force  of  772  foot-pounds  is  expended  in  heating  a 
pound  of  water  1°  F.,  therefore  to  heat  the  same  from  32°  to  212° 


FIG.  311. 


ISOTHERMAL    LINES.  345 

requires  a  force  of  138,960  foot-pounds  ;  and  to  change  the  same 
pound  of  water  into  steam  of  atmospheric  tension  requires  an 
additional  force  of  746,900  foot-pounds  (Art.  517). 

The  efficiency  of  the  best  types  of  condensing  engines  does  not 
exceed  about  65  per  cent,  of  the  work  of  the  steam  delivered  to 
them  ;  while  the  engine  and  boiler  combined  do  not  realize  more 
than  20  per  cent,  of  the  energy  of  the  fuel  consumed. 


CHAPTER    V. 

TEMPERATURE  OF  THE  ATMOSPHERE.— MOISTURE  OF  THE 
ATMOSPHERE.— DRAFT  AND  VENTILATION. 

542.  Manner  in  which  the  Air  is  Warmed. — The  space 
through  which  the  earth  moves  around  the  sun  is  intensely  cold, 
probably  75°  below  zero ;   and  the  one  or  two  hundred  miles  of 
height  occupied  by  the  atmosphere  is  too  cold  for  animal  or  vege- 
table life,  except  the  lowest  stratum,  three  or  four  miles  in  thick- 
ness.    This  portion  receives  its  heat  mainly  by  convection.     The 
radiated  heat  of  the  sun  passes  through  the  air,  warming  it  but 
little,  and  on  reaching  the  earth  is  partly  absorbed  by  it.     The  air 
lying  in  contact  with  the  earth,  and  thus  becoming  warmed,  grows 
lighter  and  rises,  while  colder  portions  descend  and  are  warmed  in 
their  turn.     So  long  as  the  sun  is  shining  on  a  given  region  of  the 
earth,  this  circulation  is  going  on  continually.     But  the  heated 
air  which  rises  is  expanded  by  diminished  pressure,  and   thus 
cooled.    Hence  the  circulation  is  limited  to  a  very  few  miles  next 
to  the  earth. 

543.  Limit  of  Perpetual  Frost.— At  a  moderate  eleva- 
tion, even  in  the  hottest  climate,  the  temperature  of  the  air  is 
always  as  low  as  the  freezing  point.     Hence  the  permanent  snow 
on  the  higher  mountains  in  all  climates.     The  limit  at  the  equa- 
tor is  about  three  miles  high,  and,  with  many  local  exceptions,  it 
descends  each  way  to  the  polar  regions,  where  it  is  very  near  the 
earth.     The  descent  is  more  rapid  in  the  temperate  than  in  the 
torrid  or  frigid  zones. 

544.  Isothermal  Lines. — These  are  imaginary  lines  on  each 
hemisphere,  through  all  those  points  whose  mean  annual  tempera- 
ture is  the  same.     At  the  equator,  the  mean  temperature  is  about 
82°  F.,  and  it  decreases  each  way  toward  the  poles,  but  not  equally 


346  HEAT. 

on  all  meridians.  Hence  the  isothermal  lines  deviate  widely  from 
parallels  of  latitude.  Their  irregularities  are  due  to  the  difference 
between  land  and  water,  in  absorbing  and  communicating  heat, 
to  the  various  elevations  of  land,  especially  ranges  of  mountains, 
to  ocean  currents,  &c.  In  the  northern  hemisphere,  the  isother- 
mal lines,  in  passing  westward  round  the  earth,  generally  descend 
toward  the  equator  in  crossing  the  oceans,  and  ascend  again  in 
crossing  the  continents.  For  example,  the  isothermal  of  50°  R, 
which  passes  through  China  on  the  parallel  of  44°,  ascends  in 
crossing  the  eastern  continent,  and  strikes  Brussels,  lat.  51°  ;  and 
then  on  the  Atlantic,  descends  to  Boston,  lat.  42°,  whence  it  once 
more  ascends  to  the  N.  W.  coast  of  America.  The  lowest  mean 
temperature  in  the  northern  hemisphere  is  not  far  from  zero,  but 
it  is  not  situated  at  the  north  pole.  Instead  of  this,  there  are  two 
poles  of  greatest  cold,  one  on  the  eastern  continent,  the  other  on 
the  western,  near  20°  from  the  geographical  pole.  There  are  in- 
dications, also,  of  two  south  poles  of  maximum  cold. 

645.  Moisture  of  the  Atmosphere.— By  the  heat  of  the 
sun  all  the  waters  of  the  earth  form  above  them  an  atmosphere  of 
vapor,  or  invisible  moisture,  having  more  or  less  extent  and  ten- 
sion, according  to  several  circumstances.  Even  ice  and  snow,  at 
the  lowest  temperatures,  throw  off  some  vapor. 

At  a  given  temperature,  there  can  exist  an  atmosphere  of 
vapor  of  the  same  height  and  tension,  whether  there  is  an  atmos- 
phere of  oxygen  and  nitrogen  or  not  (Art.  528).  Vapor  is  not 
suspended  in  the  air,  or  dissolved  by  it,  but  exists  independently. 
And  yet  it  is  by  no  means  always  true  that  there  is  actually  the 
same  tension  of  vapor  as  there  would  be  if  it  existed  alone,  because 
of  the  time  required  for  the  formation  of  vapor,  on  account  of 
mechanical  obstruction  presented  by  the  air  ;  whereas,  if  no  air 
existed,  the  vapor  would  form  almost  instantly. 

546.  Temperature  and  Tension  of  Vapor. — The  degree 
of  tension  of  vapor  forming  without  obstruction,  depends  on  its 
temperature,  but  varies  far  more  rapidly,  increasing  pretty  nearly 
in  a  geometrical  ratio,  while  the  heat  increases  arithmetically 
(Art.  534) ;  thus  the  tension  at  212°  F.  is  1  atmosphere,  at  249°  F. 
is  2  atmospheres,  at  291°  F.  is  4  atmospheres,  and  at  339°  F.  is  8 
atmospheres.  Hence,  if  vapor  should  receive  its  full  increment 
of  tension,  while  the  thermometer  rises  10  degrees  from  80°  to 
90°,  a  vastly  greater  quantity  would  be  added  than  when  it  rises 
10  degrees  from  40°  to  50°.  On  the  contrary,  if  vapor  is  at  its 
full  tension  in  each  case,  much  more  water  will  be  precipitated  in 
cooling  from  90°  to  80°  than  from  50°  to  40°. 


HYGROMETERS.  347 

547.  Dew-point. — This  is  the  temperature  at  which  vapor, 
in  a  given  case,  is  precipitated  into  water  in  some  of  its  forms. 
If  there  was  no  air,  the  dew-point  would  always  be  the  same  as 
the  existing  temperature ;  since  lowering  the  temperature  in  the 
least  degree  would  require  a  diminished  tension  or  quantity  of 
vapor,  some  must  therefore  be  condensed  into  water.     But  in  the 
air  the  tension  may  not  be  at  its  full  height,  and  therefore  the 
temperature  may  need  to  be  reduced  several  degrees  before  pre- 
cipitation will  take  place.     A  comparison  of  the  temperature  with 
the  dew-point  is  one  of  the  methods  employed  for  measuring  the 
humidity  of  the  air. 

548.  Measure  of  Vapor. — The  measure  of  the  vapor  exist- 
ing at  a* given  time,  is  expressed  by  two  numbers,  one  indicating 
its  tension, — i.  e.,  the  height  of  the  column  of  mercury  which  it 
will  sustain;  the  other,  humidity, — i.e.,  its  quantity  per  cent.,  as 
compared  with  the  greatest  possible  amount  at  that  temperature. 
Thus,  tension  =  0.6,  humidity  =  83,  signifies  that  the  quantity  of 
vapor  is  sufficient  to  support  six-tenths  of  an  inch  of  mercury,  and 
is  83  hundredths  of  the  quantity  which  could  exist  at  that  tem- 
perature.   The  greatest  tension  possible  at  zero  F.,  is  0.04  ;  at  the 
freezing  point,  0. 18  ;  at  80°  F.,  1. 0.    At  the  lowest  natural  tempera- 
tures, the  maximum  tension  is  doubled  every  12°  or  14° ;  at  the 
highest,  every  21°  or  22°. 

549.  Hygrometers.  —  This  is  the  name  usually  given  to 
instruments  intended  for  measuring  the  moisture  of  the  air.    But 
the  one  most  used  of  late  years  is  called  the  psychrometer,  which 
gives  indication  of  the  amount  of  moisture  by  the  degree  of  cold 
produced  in  evaporation ;  for  evaporation  is  more  rapid,  and  there- 
fore the  cold  occasioned  by  it  the  greater,  according  as  the  air 
is  drier.      The  psychrometer  consists  of  two  thermometers,  one 
having  its  bulb  covered  with  muslin,  which  is  kept  moistened  by 
the  capillary  action  of  a  string  dipping  in  water. 

The  wet-bulb  thermometer  will  ordinarily  indicate  a  lower 
temperature  than  the  dry-bulb;  if,  in  a  given  case,  they  read 
alike,  the  humidity  is  100.  The  instrument  is  accompanied  by 
tables,  giving  tension  and  humidity  for  any  observation. 

Various  formulae  and  complete  tables  may  be  found  in  the 
"Smithsonian  Meteorological  and  Physical  Tables." 

550. — Dew. — Frost. — The  deposition  called  dew  takes  place 
on  the  surface  of  bodies,  by  which  the  air  is  cooled  below  its 
dew-point.  It  is  at  first  in  the  form  of  very  small  drops,  which 
unite  and  enlarge  as  the  process  goes  on.  Dew  is  formed  in  the 
evening  or  night,  when  the  surfaces  of  bodies  exposed  to  the  sky 


348  HEAT. 

become  cold  by  radiation.  As  soon  as  their  temperature  has  de- 
scended to  the  dew-point,  the  stratum  of  air  contiguous  to  them 
deposits  moisture,  and  continues  to  do  so  more  and  more  as  the 
cold  increases. 

Of  two  bodies  in  the  same  situation,  that  will  receive  most  dew 
which  radiates  most  rapidly.  Many  vegetable  leaves  are  good 
radiators,  and  receive  much  dew.  Polished  metal  is  a  poor  radia- 
tor, and  ordinarily  has  no  dew  deposited, on  it. 

Sometimes,  however,  good  radiators  have  little  dew.  because  they 
are  so  situated  as  to  obtain  heat  nearly  as  fast  as  they  radiate  it. 
Dew  is  rarely  formed  on  a  bed  of  sand,  though  it  is  a  good  radia- 
tor, because  the  upper  surface  gets  heat  by  conduction  from  the 
mass  below.  Dew  is  not  formed  on  water,  because  the  upper 
stratum  sinks  and  gives  place  to  warmer  ones. 

Bodies  most  exposed  to  the  open  sky,  other  things  being  equal, 
have  most  dew  precipitated  on  them.  This  is  owing  to  the  fact, 
that  in  such  circumstances,  they  have  no  return  of  heat  either  by 
reflection  or  radiation.  If  a  body  radiates  its  heat  to  a  building, 
a  tree,  or  a  cloud,  it  also  gets  some  in  return,  both  reflected  and 
radiated.  Hence,  little  dew  is  to  be  expected  in  a  cloudy  night, 
or  on  objects  surrounded  by  high  trees  and  buildings. 

Wind  is  unfavorable  to  the  formation  of  dew,  because  it  mingles 
the  strata,  and  prevents  the  same  mass  from  resting  long  enough 
on  the  cold  body  to  be  cooled  down  to  the  dew-point. 

When  the  radiating  body  is  cooled  below  the  freezing  point, 
the  water  deposited  takes  the  solid  form  in  fine  crystals,  and  is 
called  frost.  Frost  will  often  be  found  on  the  best  radiators,  or 
those  exposed  to  the  open  sky,  when  only  dew  is  found  elsewhere. 

551.  Fog". — This  form  of  precipitation  consists  of  very  small 
globules  of  water  sustained  in  the  lower  strata  of  the  air.  Fog 
occurs  most  frequently  over  low  grounds  and  bodies  of  water, 
where  the  humidity  is  likely  to  be  great.  If  air  thus  humid  mixes 
with  air  cooled  by  neighboring  land,  even  of  less  humidity,  there 
will  probably  be  more  vapor  than  can  exist  at  the  intermediate 
temperature,  for  the  reason  mentioned  in  Art.  546.  The  case  may 
be  illustrated  thus.  Let  two  masses  of  air  of  equal  volumes  be 
mixed,  the  temperature  of  one  being  40°  F.,  the  other  60°  F.,  and  each 
•containing  vapor  at  the  highest  tension.  Then  the  mixture  will 
have  the  mean  temperature  of  50°,  and  the  vapor  of  the  mixture 
will  also  be  the  arithmetical  mean  between  that  of  the  two  masses. 
But,  according  to  the  law  (Art.  546),  the  vapor  can  only  have  a 
tension  which  is  nearly  a  geometrical  mean  between  the  two,  and 
that  is  necessarily  lower  than  the  arithmetical  mean  ;  hence  the 
excess  must  be  precipitated.  If  8  Ibs.  of  vapor  were  in  one  volume 


CLASSIFICATION    OF    CLOUDS.  349 

and  18  Ibs.  in  the  other,  an  equal  volume  of  the  mixture  would 
have  J  (8  +  18)  =.  13  Ibs.  of  moisture ;  but  at  the  mean  tempera- 
ture of  50°,  only  */  8  x  18  —  12  Ibs.  could  exist  as  vapor  ;  there- 
fore one  pound  must  be  precipitated. ,  And  even  if  one  of  the 
masses  had  a  humidity  somewhat  below  100,  still  some  precipita- 
tion is  likely  to  take  place. 

552.  Cloud. — The  same  as  fog,  except  at  a  greater  elevation. 
Air  rising  from  heated  places  on  the  earth,  and  carrying  vapor 
with  it,  is  likely  to  meet  with  masses  much  colder  than  itself,  and 
depositions  of  moisture  are  therefore  likely  to  take  place.    Moun- 
tain-tops are  often  capped  with  clouds,  when  all  around  is  clear. 
This   happens   when   lower  and  warmer  strata  are  driven   over 
them,  and  thus  cooled  below  the  dew-point.     The  same  air,  as  it 
continues  down  the  other  side,  takes  up  its  vapor  again,  and  is  as 
transparent  as  it  was  before  ascending.     A  person  on  the  summit 
perceives  a  chilly  fog  driving  by  him,  but  the  fog  was  an  in- 
visible vapor  a  few  minutes  before  reaching  him,  and  returns  to 
the  same  condition  soon  after  leaving  him.     The  cloud  rests  on 
the  mountain  ;  but  all  the  particles  which  compose  it  are  swiftly 
crossing  over.    Clouds  are  often  above  the  limit  of  perpetual  frost ; 
they  then  consist  of  crystals  of  ice. 

553.  Classification  of  Clouds. — The  aspects  of  clouds  are 
various,  and  depend  in  some  measure  at  least  on  the  circumstances 
of  their  formation.     The  usual  classification  is  the  following : 

1.  Cirrus. — This  cloud  is  fibrous  in  its  appearance,  like  Jtair 
or  flax',  sometimes  straight,  sometimes  bent,  and  frequently  at  one 
end  is  gathered  into  a  confused  heap  of  fibres.    The  cirrus  is  high, 
and  often  consists  of  frozen  particles,  even  in  summer. 

2.  Cumulus. — This  consists  of  compact  rounded  heaps,  which 
often  resemble  mountain-tops  covered  with  snow.     This  form  of 
cloud  is  confined  mostly  to  the  summer  season  ;  it  usually  begins 
to  form  after  the  suu  rises,  and  to  disappear  before  it  sets,  and  is 
rarely  seen  far  from  land.     The  cumulus  is  generally  not  so  high 
as  the  cirrus. 

3.  Stratus. — Sheets  or  stripes  of  cloud,  sometimes  overspread- 
ing the  whole  sky,  or  as  a  fog  covering  the  surface  of  the  earth  or 
water.     The  stratus  is  the  most  common,  and  usually  lies  lowest 
in  the  air. 

4.  5,  6.  Cirro-cumulus,  cirro-stratus,  cumulo- stratus. — Inter- 
mediate or  combined  forms. 

7.   Nimbus. — A  cloud,  which  forms  so  fast  as  to  fall  in  rain  or 
snow,  is  called  by  this  name. 

554.  Rain,  Mist. — Whether  the  precipitated  moisture  has  the 


350  HEAT. 

form  of  cloud  or  rain,  depends  on  the  rapidity  with  which  precipi- 
tation takes  place.  If  currents  of  air  are  in  rapid  motion,  if  the 
temperature  of  masses,  brought  into  contact  by  this  motion,  are 
widely  different,  and  if  their  humidity  is  at  a  high  point,  the  vapor 
will  be  precipitated  so  rapidly,  that  the  globules  will  touch  each 
other,  and  unite  into  larger  drops,  which  cannot  be  sustained. 
Globules  of  fog  and  cloud,  however,  are  specifically  as  heavy  as 
drops  of  rain  ;  but  they  are  sustained  by  the  slightest  upward 
movements  of  the  air,  because  they  have  a  great  surface  compared 
with  their  weight.  A  globule  whose  diameter  is  100  times  less 
than  that  of  a  drop  of  rain,  meets  with  100  times  more  obstruc- 
tion in  descending,  since  the  weight  is  diminished  a  million  times 
(rb-)3>  an(*  tne  surface  only  ten  thousand  times  (TJ¥)2.  So  the 
dust  of  even  heavy  minerals  is  sustained  in  the  air  for  some  time, 
when  the  same  substances,  in  the  form  of  sand,  or  coarse  gravel, 
fall  instantly. 

If  a  cloud  of  fine  dust  contains  so  much  matter  as  to  make 
the  mass  of  a  cubic  foot  of  the  dusty  air  greater  than  that  of  a 
cubic  foot  of  pure  air,  it  will  descend.  If  the  mean  density  of  a 
fog  is  greater  than  that  of  the  purer  surrounding  air,  it  will  settle 
down  into  hollows  and  valleys  ;  if  its  mean  density  is  less  than 
the  air,  it  will  rise  as  cloud. 

Mist  is  fine  rain ;  the  drops  are  barely  large  enough  to  make 
their  way  slowly  to  the  earth. 

655.  Hail,  Sleet,  Snow. — When  the  air  in  which  rapid  pre- 
cipitation occurs,  is  so  cold  as  to  freeze  the  drops,  hail  is  produced. 
As  hailstones'  are  not  usually  in  the  spherical  form  when  they 
reach  the  earth,  it  is  supposed  that  they  are  continually  receiving 
irregular  accretions  in  their  descent  through  the  vapor  of  the  air. 
Hail-storms  are  most  frequent  and  violent  in  those  regions  where 
hot  and  cold  bodies  of  air  are  most  easily  mixed.  Such  mixtures 
are  rarely  formed  in  the  torrid  zone,  since  there  the  cold  air  is  at  a 
great  elevation  ;  in  the  frigid  zone,  no  hot  air  exists  at  any  height ; 
but  in  the  temperate  climates,  the  heated  air  of  the  torrid,  and  the 
intensely  cold  winds  of  the  frigid  zone,  may  be  much  more  easily 
brought  together ;  and  accordingly,  in  the  temperate  zones  it  is 
that  hail-storms  chiefly  occur.  Even  in  these  climates,  they  are  not 
frequent  except  on  plains  and  in  valleys  contiguous  to  mountains 
which  are  covered  with  snow  during  the  summer.  The  slopes  of 
the  mountain  sides  give  direction  to  currents  of  air,  so  that  masses 
of  different  temperature  are  readily  mingled  together. 

Sleet  is  frozen  mist,  that  is,  it  consists  of  very  small  hailstones. 

Snow  consists  of  the  small  crystals  of  frozen  cloud,  united  in 
flakes.  Like  all  transparent  substances,  when  in  a  pulverized 


CYCLONES.  351 

state,  it  owes  its  whiteness  to  innumerable  reflecting  surfaces.  A 
cloud,  when  the  sun  shines  upon  it,  is  for  the  same  reason  in- 
tensely white  (Art.  371). 

556.  Theories  of  Precipitation. — It  is  probable  that  clouds 
and  rain  are  caused  not  only  by  the  mixing  of  air  of  different  tem- 
peratures, but  also  by  the  changes  which  take  place  in  the  condi- 
tion of  the  air  as  it  ascends. 

In  the  lower  strata,  the  air  is  about  one  degree  colder  for  every 
300  feet  of  elevation.  If,  therefore,  a  mass  of  air  is  transferred 
from  the  surface  of  the  earth  to  a  height  in  the  atmosphere,  it  will 
be  cooled  to  the  temperature  of  the  stratum  which  it  reaches;  not 
principally  by  giving  off  its  heat,  but  by  expanding,  and  thus  hav- 
ing its  own  heat  reduced  by  being  diffused  through  a  larger  space. 
Now,  if  the  rising  mass  was  saturated  with  moisture,  this  moist- 
ure would  begin  at  once  to  be  precipitated  by  the  cooling  which 
it  undergoes  in  consequence  of  expansion.  If,  instead  of  being 
saturated,  its  dew-point  is  a  certain  number  of  degrees  below  its 
temperature,  it  must  ascend  far  enough  to  be  cooled  to  the  dew- 
point,  before  precipitation  of  its  moisture  will  take  place.  Sup- 
pose, for  instance,  the  temperature  at  the  earth  is  70°  F.,  and  the 
dew-point  is  65°  F. ;  then  after  the  warm  air  has  risen  1500  feet 
(5  x  300  ft.),  it  will  become  5°  cooler,  and  contain  all  the  moisture 
which  is  possible  at  that  temperature.  At  that  point  precipitation 
begins,  and  forms  the  base  of  a  cloud.  The  clouds,  called  cumulus, 
which  are  seen  forming  during  many  summer  forenoons,  are  the 
precipitations  of  columns  rising  from  warm  spots  of  earth  so  high 
that  they  are  cooled  below  their  dew-point.  But  the  movement 
and  the  precipitation  do  not  stop  here  ;  for,  as  moisture  is  pre- 
cipitated, its  latent  heat  is  given  off  in  large  quantities,  which 
elevates  the  temperature  of  the  mass,  and  causes  it  to  rise  still 
higher,  and  precipitate  still  more  of  its  moisture.  As  it  becomes 
rarer,  it  spreads  laterally,  and  causes  the  cumulus  often  to  assume 
the  overhanging  form  which  distinguishes  that  species  of  cloud. 

557.  Cyclones.  —  The   late  Mr.  Redfield  investigated  with 
great  success  the  phenomena  of  violent  storms,  especially  of  At- 
lantic hurricanes,  and  showed   that  they  are  generally,   if  not 
always,  great  whirlwinds,  called  cyclones.     They  usually  take  their 
rise  in  the  equatorial  region  eastward  of  the  West  India  Islands ; 
they  rotate  on  a  vertical  axis,  advancing  slowly  to  the  northwest, 
until  they  approach  the  coast  of  the  United  States  near  the  lati- 
tude of  30°,  and  then  gradually  veer  to  the  northeast,  running 
nearly  parallel  to  the  American  coast,  and  finally  spend  themselves 
in  the  northern  Atlantic.     Their  rotary  motion  is  always  in  one 


352  HEAT. 

direction,  namely,  from  the  east  through  the  north  to  the  west, 
or  against  the  sun.  This  motion  is  also  far  more  violent,  especially 
in  the  central  parts  of  the  storm,  than  the  progressive  motion.. 
The  rotary  motion  may  amount  to  50  or  100  miles  per  hour, 
while  the  forward  motion  of  the  storm  is  not  more  than  15  or  2Q> 
miles. 

In  the  southern  hemisphere  also,  cyclones  occur,  having  a 
progressive  and  a  rotary  motion,  both  symmetrical  with  those  of 
the  northern  cyclones.  On  the  axis  they  revolve  with  the  sun, 
not  against  it ;  and  they  first  advance  toward  the  southwest,  and 
gradually  veer  toward  the  southeast,  as  they  recede  from  the 
equator. 

558.  Draught  of  Flues. — The  effect  of  the  sun's  heat  in 
causing  circulation  of  the  air  has  been  already  considered  (Art. 
268-272).  Similar  movements  on  a  limited  scale  are  produced 
whenever  a  portion  of  the  air  is  heated  by  artificial  means.  Thus, 
the  air  of  a  chimney  is  made  lighter  by  a  fire  beneath  it,  than  a 
column  of  the  outer  air  extending  to  the  same  height.  It  is  there- 
fore pressed  upward  by  the  heavier  external  air,  which  descends 
and  moves  toward  the  place  of  heat.  The  difference  of  weight  in 
the  two  columns  is  greater,  and  therefore  the  draught  stronger, 
if  the  chimney  is  high,  provided  the  supply  of  heat  is  sufficient 
to  maintain  the  requisite  temperature.  Chimneys  are  frequently 
built  one  or  two  hundred  feet  high  for  the  uses  of  manufactories. 
The  high  fireplaces  and  large  flues  of  former  times  were  un- 
favorable for  draught,  both  because  much  cold  air  could  mingle 
with  that  which  was  heated,  and  because  there  was  room  for 
external  air  to  descend  by  the  side  of  the  ascending  column.  For 
good  draught,  no  air  should  be  allowed  to  enter  the  flue  except 
that  which  has  passed  through  the  fire. 

559.  Ventilation  of  Apartments. — The  air  of  an  apart- 
ment, as  it  becomes  vitiated  by  respiration,  may  generally  be 
removed,  and  fresh  air  substituted,  by  taking  advantage  of  the 
same  inequality  of  weight  in  air-columns,  which  has  been  men- 
tioned. If  opportunity  is  given  for  the  warm  impure  air  to  escape 
from  the  top  of  a  room,  and  for  external  air  to  take  its  place,  there 
will  be  a  constant  movement  through  the  room,  as  in  the  flue  of  a 
chimney,  though  at  a  slower  rate.  If  the  external  air  is  cold, 
the  weight  of  the  columns  differs  more,  and  therefore  the  ventila- 
tion is  more  easily  effected.  But  in  cold  weather,  the  air,  before 
being  admitted  to  the  room,  is  warmed  by  passing  through  the 
air-chambers  of  a  furnace.  When  there  is  a  chimney-flue  in  the 
wall  of  a  room,  with  a  current  of  hot  air  ascending  in  it,  the 
ventilation  is  best  accomplished  by  admitting  the  air  into  the-- 


SOURCES    OF    HEAT. 


353 


FIG.  312. 


flue  at  the  upper  part  of  the  room  ;  since  it  will  then  be  removed 
with  the  velocity  of  the  hot-air  current. 

The  tendency  of  the  air  of  a  warm  room  to  pass  out  near  the 
top,  while  a  new  supply  enters  at  the  lower  part,  is  shown  by 
holding  the  flame  of  a  candle  at  the  top,  and  then  at  the  bottom, 
of  a  door  which  is  opened  a  little  distance.  The  flame  bends  out- 
ward at  the  top  and  inward  at  the  bottom. 

The  impure  air  of  a  large  audience-room  is  sometimes  removed 
by  a  mechanical  contrivance,  as,  for  instance,  a  fan-wheel  placed 
above  an  opening  at  the  top,  and  driven  by  steam. 

The  ventilation  of  mines  is  accomplished  sometimes  by  a  fire 
built  under  a  shaft,  fresh  air  being  supplied  by  another  shaft,  and 
sometimes  by  a  fan- wheel  at  the  top  of  the  shaft.  If  there  happen 
to  be  two  shafts  which  open  to  the  surface  at  very  different  eleva- 
tions, ventilation  may  be  effected  by  the  inequality  of  temperature 
which  is  likely  to  exist  within  the  earth  and  above  it.  Let  M  M 
(Fig.  312)  be  the  vertical  section  of  a  mine  through  two  shafts  A 
and  B,  which  open 
at  different  heights 
to  the  surface  of  the 
earth.  If  the  exter- 
nal air  is  of  the  same 
temperature  as  the 
air  within  the  earth, 
then  the  column  A 
in  the  longer  shaft 
has  the  same  weight 
as  j£and  C  together, 
measured  upward  to 
the  same  level.  In 
that  case,  which  is 
likely  to  occur  in 

spring  and  fall,  there  is  no  circulation  without  the  use  of 
means.  But  in  summer  the  air  C  is  warmer  than  A  and  B  ;  there- 
fore A  is  heavier  than  B  4-  C.  Hence  there  is  a  current  of  air 
down  A  and  up  B.  In  winter,  C  is  colder  than  air  within  the 
earth  ;  therefore  B  4-  C  are  together  heavier  than  A,  and  the 
current  sets  in  the  opposite  direction,  down  B  and  up  A. 

560.  Sources  of  Heat. — The  sun,  although  nearly  a  hun- 
dred millions  of  miles  from  the  earth,  is  the  source  of  nearly  all 
the  heat  existing  at  its  surface.  The  interior  of  the  earth,  except 
a  thickness  of  forty  or  fifty  miles  next  to  the  surface,  is  believed  to 
be  in  a  condition  of  heat  so  intense  that  all  the  materials  compos- 
ing it  are  in  the  melted  state.  But  the  earth's  crust  is  so  poor  a 
23 


354  HEAT. 

conductor  that  only  an  insensible  fraction  of  all  this  heat  reaches 
the  surface. 

Mechanical  operations  are  usually  attended  by  a  development 
of  heat.  For  example,  if  a  broad  surface  of  iron  were  made  to 
revolve,  rubbing  against  another  surface,  nearly  all  the  force  ex- 
pended in  overcoming  the  friction  would  appear  as  heat,  a  com- 
paratively small  part  being  conveyed  through  the  air  as  sound. 
The  cutting  tool  employed  in  turning  an  iron  shaft  has  been 
known  to  generate  heat  enough  to  raise  a  large  quantity  of  cold 
water  to  the  boiling  point,  and  to  keep  it  boiling  for  an  indefinite 
time.  It  is  a  fact  familiar  to  all,  that  violent  friction  of  bodies 
against  each  other  will  set  combustibles  on  fire.  The  axles  of 
railroad  cars  are  made  red-hot  if  not  duly  oiled  ;  boats  are  set  on 
fire  by  the  rope  drawn  swiftly  over  the  edge  by  a  whale  after  he 

is  harpooned  ;  a  stream  of 
sparks  flies  from  the  emery 
wheel  when  steel  is  polished, 
&c. 

A  lecture  illustration  de- 
vised by  Tyndall  will  show 
the  conversion  of  motion 
into  heat. 

Screw  a  brass  tube  A, 
about  4  inches  long  and  J 
inch  in  diameter,  upon  the 

spindle  of  a  whirling  table  B  (Fig.  313).  Nearly  fill  the  tube 
with  water  and  insert  a  cork  ;  press  the  tube  between  the  jaws  of 
a  wooden  clamp  (7,  while  A  is  rapidly  rotating.  Heat 
is  developed  by  the  friction,  and  this  communicated 
to  the  water  causes  it  to  boil,  and  finally  to  eject  the 
cork. 

The  heat  developed  by  sudden  compression  of  air 
may  be  rendered  visible  by  igniting  vapor  of  carbon 
bisulphide  in  a  "  Fire  Syringe."  A  thick  glass  tube  A 
(Fig.  314)  closed  at  the  lower  end,  has  a  well-fitted 
piston  e,  whose  rod  is  terminated  by  a  wide  cap,  or 
button,  B,  upon  which  the  palm  of  the  hand  may 
strike  forcibly  without  injury.  If  a  bit  of  tinder,  or  a 
small  tuft  of  cotton  moistened  with  carbon  bisulphide, 
be  placed  at  the  bottom  of  the  tube,  it  will  be  ignited 
when  the  piston  is  driven  down  by  a  sudden  blow  upon 
it. 

The  heat  due  to  percussion  may  be  found  thus  : 
Let  w  (  —  10  Ibs.)  be  the  weight  of  a  lead  ball  and 


CHEMICAL    ACTION.  355 

v  (  =  1000  ft.  per  sec.),  be  its  velocity  ;   then  its  energy  in  foot- 

W  v2 

pounds  is  (Art  3?)  -  —  =  155279  foot-pounds.     Since  772  foot- 
*  9 


pounds  is  the  equivalent  of  one  thermal  unit  (F)  we  find 
=  201  thermal  units,  to  be  the  heat  developed  by  the  sudden 
stopping  of  the  ball.  Suppose  none  of  this  heat  to  be  transmitted 
to  the  body  struck  ;  then  taking  the  specific  heat  of  lead  as  .0314 
(Art.  513),  we  find  the  temperature  of  the  ball,  assuming  its  origi- 
nal temperature  to  be  60°  F.,  to  be 


a  temperature  above  that  of  the  melting  point  of  lead,  which  is 
635°  F. 

Indeed,  luherever  the  full  equivalent  of  any  force  is  not  obtained 
in  some  other  form,  the  deficiency  may  be  detected  in  the  heat  which 
is  developed. 

Chemical  action  is  another  very  common  source  of  heat.  Com- 
bustion is  the  effect  of  violent  chemical  attraction  between  atoms 
of  different  natures,  when  both  light  and  heat  are  manifested.  If 
the  union  goes  on  slowly,  as  in  the  rusting  of  iron,  the  amount  of 
heat  is  the  same,  but  it  is  diffused  as  fast  as  developed.  The 
molecular  forces,  expended  in  most  cases  of  chemical  combination, 
as  measured  by  their  heating  effects,  are  enormously  great. 

The  warmth  produced  by  the  vital  processes  in  plants  and  ani- 
mals is  supposed  by  many  physicists  to  be  caused  by  chemical 
action.  In  breathing  the  air,  some  of  its  oxygen  is  consumed, 
which  becomes  united  with  the  blood.  This  process  is  in  some 
respects  analogous  to  a  slow  combustion,  by  which  heat  is  evolved 
in  the  animal  system. 


PART    VII, 


CHAPTER    I. 

THE    MAGNET    AND   ITS    PROPERTIES. 

661.  The  Magnet.— Fragments  of  iron  ore  are  sometimes 
found  which  strongly  attract  iron,  and  bars  of  steel  are  artificially 
prepared  which  exhibit  the  same  property.  These  bodies  are 
called  magnets ;  the  ore  is  the  natural  magnet,  commonly  called 
lodestone  ;  the  prepared  steel  bar  is  an  artificial  magnet. 

Ores  of  iron  which  are  attracted  by  the  magnet  are  abundant, 
both  massive  and  in  grains  constituting  magnetic  iron  sand. 

562.  The  Attraction  between  a  Magnet  and  Iron.— 
The  magnetic  property  which  is  likely  to  be  first  noticed  is  the 
attraction  of  iron.  If  a  lodestone  or  a  bar  magnet  be  rolled  in 
iron  filings  (Fig.  315),  there  are  two  opposite  points  to  which  the 

FIG.  315. 


filings  attach  themselves  in  thick  clusters,  arranged  in  diverging 
filaments.  These  opposite  points  of  greatest  action  are  called 
poles.  The  straight  line  joining  these  poles  is  called  the  axis  of 
the  magnet. 

The  mutual  attraction  between  a  magnet  and  iron  is  shown 
by  bringing  a  piece  of  iron  toward  either  pole  of  the  magnetic 
needle  ;  the  needle  instantly  turns  so  as  to  bring  its  pole  as  near 
as  possible  to  the  iron  (Fig.  316).  On  the  other  hand,  an  iron 
needle  being  suspended  in  like  manner,  the  same  movement  takes 
place,  when  either  pole  of  a  magnet  is  brought  near  to  it. 


POLARITY. 


357 


FIG.  316. 


That  this  attraction  diminishes  from  the  poles  towards  the 

middle  point  of  the  axis  is 

indicated  by  the  arrange- 
ment of  the  iron  filings  in 

Fig.  315.     The  variation  in 

the  force  may  be  illustrated 

by  means  of  a  light  balance 

beam,    from    one    end    of 

which  is  suspended  a  short 

piece  of  soft  iron  wire  as  in 

Fig.  317,  counterpoised  by 

weights  in  the  scale-pan.  As  the  bar  magnet  M  is  moved  along 

below  the  wire  A,  at  a 

FIG.  317. convenient  distance  from 

it,  the  weights  required 
to  keep  the  beam  hori- 
zontal indicate  the  varia- 
tions in  the  force. 

563.  Polarity.— If  a 

light  magnet  is  delicately 
suspended  on  a  pivot  at 
the  neutral  point,  as  in 
Fig.  318,  it  is  called  a 

magnetic  needle.  When  thus  placed  and  left  to  itself,  it  oscillates 
for  a  time,  and  finally  settles  with  its  axis  in  a  certain  fixed 
direction,  which  in  most  places  is  nearly 
north  and  south.  The  end  which  points 
in  a  northerly  direction  is  called  the 
north  pole ;  the  other,  the  south  pole. 
These  poles  are  usually  marked  on  the 
larger  magnets  by  the  letters  J^and  S, 
so  that  they  may  be  instantly  distin- 
guished. If  a  magnetic  needle  has  simply  a  mark  or  stain  on  one 
end,  that  end  is  understood  to  be  the  north  pole. 

Instead  of  using  the  terms  north  pole  and  south  pole  it  is  better 
to  speak  of  the  marked  pole  and  unmarked  pole,  to  prevent  con- 
fusion when  discussing  terrestrial  magnetism.  The  pole  is  not  at 
the  extremity  of  the  needle,  but  at  a  little  distance  from  the  end, 
and  is  not  to  be  regarded  as  a  point  but  rather  ;;s  a  resultant 
centre  of  forces. 

564.  Action  of  Magnets  on  Each  Other.— While  either 
pole  of  a  magnet  attracts  and  is  attracted  by  a  piece  of  iron,  it  is 
otherwise  when  the  pole  of  one  magnet  is  brought  near  the  pole 


FIG.  318. 


358  MAGNETISM. 

of  another.     There  is  attraction  in  some  cases,  and  repulsion  in 

others.     If  the  magnets  are  properly 
FlG-  319-  marked,  and  one  of  them  suspended 

so  as  to  move  freely,  it  is  readily 
discovered  that  the  law  of  action  is 
the  following  : 

Poles  of  the  same  name  repel,  and 
those  of  contrary  name  attract  each 
other. 

Thus,  the  pole  S  of  the  magnet 
(Fig.  319)  repels  s  of  the  needle,  and 
attracts  n  ;  and  if  the  magnet  were 
inverted,  and  the  pole  N  brought  near  to  n,  the  latter  would  be 
repelled,  and  s  be  attracted. 

565.  Magnets    and    Magnetic    Substances. — All  sub- 
stances which  are  attracted  by  the  magnet  are  classed  as  magnetic 
substances.     Among  these  iron,  nickel,  cobalt,  chromium,  and 
manganese  are  the  most  marked. 

Magnetic  substances  show  no  polarity  and  exert  no  influence 
upon  each  other. 

Magnets,  on  the  contrary,  manifest  polarity  and  attract  all 
magnetic  substances,  and  attract  or  repel  other  magnets.  Perma- 
nent magnets  may  be  made  of  steel,  cast-iron,  nickel,  and  cobalt, 
the  magnetism  of  the  last  being  very  feeble. 

To  determine  whether  a  bar  is  a  magnet,  or  merely  magnetic, 
repulsion  is  a  surer  test  than  attraction ;  for  a  magnet  will  attract 
any  magnetic  substance,  but  repulsion  can  only  be  due  to  similar 
polarities. 

To  find  experimentally  whether  a  body  is  magnetic,  lay  a  piece 
of  stiff  paper  upon  the  end  of  a  powerful  bar-magnet  held  verti- 
cally, and  upon   the   paper  place  a  fine,   short 
needle,  the  eye-end  having  been  broken  off,   so       FIG.    820. 
that  it  may  rest  upon  its  point  as  in  Fig.  320,  in 
which  JVis  the  marked  pole  of  the  magnet,  cd 
the  paper  or  card,  and  n  the  needle.     Move  the 
paper  until  a  point  on  the  face  of  the  magnet  is 
found  at  which  the  needle  will  stand  nearly  verti- 
cal, and  then  raise  the  paper  as  far  as  possible 
without  causing  the  needle  to  incline,  when  it 
will  be  very  sensitive  to  magnetic  influence.     Any 
substance  having  even  a  trace  of  magnetic  force,  will  cause  the 
needle  to  sway  aside  on  being  brought  near  its  upper  end. 

566.  Magnetic  Induction. — When  a  bar  of  iron  is  brought 


TP 


MAGNETIC    INDUCTION.  359 

near  to  the  pole  of  a  magnet,  though  attraction  is  the  phenome- 
non first  observed,  as  stated  (Art.  562),  yet  it  is  readily  proved 
that  this  attraction  results  from  a  change  which  is  previously 
produced  in  the  iron.  It  becomes  a  magnet  through  the  influence 
of  the  magnet  which  is  near  it.  That  end  of  the  iron  bar  which 
is  placed  near  one  pole  of  a  magnet  becomes  a  pole  of  the  opposite 
name,  and  the  remote  end  a  pole  of  the  same  name.  Hence, 
according  to  the  law  (Art.  564),  the  poles  which  are  contiguous 
attract  each  other,  because  they  are  unlike.  The  influence  by 
which  the  iron  becomes  a  magnet  is  called  induction.  A  magnet, 
when  brought  near  to  a  piece  of  iron,  induces  upon  the  iron  the 
magnetic  condition,  without  any  loss  of  its  own  magnetic  proper- 
ties. This  influence  is  more  powerful  according  as  the  two  are 
nearer  to  each  other ;  it  is,  therefore,  greatest  when  the  two  bars 
are  in  contact. 

That  the  iron  is  truly  a  magnet  for  the  time  being  may  be 
proved  by  sifting  iron  filings  over  it  as  in  Art.  562,  when  the 
poles  and  the  neutral  line  will  be  made  apparent. 

567.  Successive  Inductions.— Let  a  bar  of  iron,  B  (Fig. 
321),  be  suspended  from  the  unmarked  pole  of  the  magnet  A ; 
then  the  upper  end  of  B  is  a  marked 
pole,   and   the   lower  end   an   un-  FIG.  321. 

marked  pole.  Now,  as  B  is  a  mag- 
net, it  will  induce  the  magnetic 
state  on  another  bar,  C,  when 
brought  in  contact ;  and,  as  before, 
the  poles  of  opposite  name  will  be 
contiguous.  Therefore,  the  upper 
end  of  C  is  marked,  and  the  lower 
end  unmarked.  D  is  also  a  magnet  by  the  inductive  power  of  C. 
Thus,  there  is  an  indefinite  series  of  inductions,  growing  weaker, 
however,  from  one  to  another,  as  the  number  is  greater,  and  as 
the  bars  are  longer. 

The  filaments  of  iron  filings  which  attach  themselves  to  the 
pole  of  a  magnet  (Art.  562)  are  so  many  series  of  small  magnets 
formed  in  the  same  manner  as  just  described.  Every  particle  of 
iron  is  a  complete  magnet,  having  its  poles  so  arranged  that  the 
opposite  poles  of  two  successive  particles  are  always  contiguous. 

568.  Reflex  Influence. — When  a  magnet  exerts  the  induc- 
tive power  upon  a  piece  of  iron  which  is  near  it,  its  own  magnetic 
intensity  is  increased.  The  end  of  the  piece  of  iron  contiguous 
to  the  pole  of  the  magnet  is  no  sooner  endued  with  the  opposite 
polarity  than  it  reacts  upon  the  magnet,  and  increases  its  inten- 


360 


MAGNETISM. 


sity ;  so  that,  if  fragments  of  iron  are  attached  to  a  magnet,  as 
many  as  it  will  sustain,  then  after  a  time  another  may  be  added, 
and  again  another,  till  there  is  a  very  sensible  increase  of  its  origi- 
nal power. 

Hence,  too,  the  force  of  attraction  of  the  dissimilar  poles  of 
two  magnets  is  greater  than  the  force  of  repulsion  of  the  simi- 
lar poles ;  because,  when  the  poles  are  unlike,  each  acts  induc- 
tively on  the  other  to  develop  its  poles,  more  fully;  but  when 
they  are  alike,  the  influence  which  they  reciprocally  exert  tends 
to  make  them  unlike,  and  of  course  to  diminish  their  repulsive 
force. 

An  extreme  case  of  this  diminution  of  repulsive  force  occurs 
when  the  like  poles  of  two  very  unequal  magnets  are  brought  into 
contact.  The  small  magnet  immediately  clings  to  the  large  one, 
as  though  the  poles  were  unlike  ;  and  if  examined,  it  is  found 
that  they  are  unlike.  The  powerful  magnet  has  in  an  instant 
reversed  the  poles  of  the  weak  one  by  its  strong  inductive  power, 
the  latter  not  having  force  enough  to  diminish  sensibly  the  strength 
of  the  other. 

569.  Double  Induction. — The  effects  of  two  inductions  at 
once  on  a  bar  of  iron  are  various. 

1.  The  bar  may  become  a  single  magnet  of  double  strength. 

2.  It  may  consist  of  two  distinct  magnets. 

3.  It  may  have  no  magnetic  power  at  all. 

FlQ  322  The  first  case  is  illustrated  by  bring- 

ing the  north  pole  of  a  magnet  to  one 
end  of  the  iron,  and  the  south  pole 
of  another  magnet  to  the  other  end. 
Each  magnet  will  form  two  poles  by 
induction,  and  it  is  evident  that  the 
two  pairs  of  poles  will  coincide.  Even 
one  magnet  produces  the  same  effect 
when  laid  by  the  side  of  a  bar  of  iron 
of  the  same  length. 

This  case  may  also  be  experiment- 
ally illustrated  by  hanging  a  bar-mag- 
net upon  a  spring  balance,  as  in  Fig. 
322,  and  noting  the  force  required  to 
detach  a  soft  iron  rod,  «,  suspended 
from  the  magnet ;  having  again  brought 
the  rod  into  contact  with  the  magnet, 
touch  the  lower  end  of  the  rod  a  with 
a  second  magnet,  equal  in  all  respects 
to  the  first,  the  poles  being  arranged 


COERCIVE    FORCE.  361 

as  in  the  figure,  and  note  the  force  required  to  detach  the  second 
magnet ;  the  latter  force  will  be  found  to  be  about  double  the 
former,  the  exactness  of  the  result  depending  upon  the  degree  of 
approximation  to  theoretically  perfect  conditions. 

To  show  the  second  effect,  apply  one  pole  of  a  magnet  to  the 
middle  of  the  iron  bar ;  then  an  opposite  pole  is  formed  at  the 
middle,  and  a  like  pole  at  each  end,  each  half  of  the  bar  being  a 
separate  magnet.  The  same  effect  is  produced  by  bringing  the 
like  poles  of  two  magnets  in  contact  with  the  ends  of  the  bar  ;  for 
both  ends  will  be  of  the  opposite  kind,  and  the  middle  of  the 
same  kind,  as  the  poles  applied.  If  a  pole  is  applied  to  the  middle 
of  a  star  of  iron,  the  extremity  of  each  ray  is  a  pole  of  the  same 
kind ;  if  to  the  middle  of  a  circle  of  iron,  the  same  polarity  is 
found  at  every  point  of  the  circumference. 

As  an  example  of  the  third  case,  suspend  a  bar  of  iron  from 
the  pole  of  a  magnet,  and  then  bring  the  opposite  pole  of  an 
equal  magnet  to  the  point  of  contact ;  the  two  poles  induced  by 
one  are  contrary  to  the  two  induced  by  the  other,  and  they  are 
found  to  be  completely  neutralized,  as  is  indicated  by  the  fall  of 
the  bar. 

This  last  case  shows  that  two  opposite  and  equal  magnetic 
poles  formed  at  the  same  point  destroy  each  other. 

570.  Coercive   Force. — If  in  the  several  experiments  on 
iron  bars,  which  have  been  already  described,  pure  annealed  iron 
is  used,  the  iron  rapidly  acquires  its  maximum  of  magnetic  inten- 
sity, and  on  being  removed  from  the  magnet  its  induced  magnet- 
ism as  rapidly  disappears.     In  ordinary  soft  iron  bars  a  trace  of 
magnetism  will  remain  for  hours,  or  perhaps  until  a  new  induction 
of  opposite  magnetism  destroys  it.     The  magnetism  so  retained  is 
called  residual  magnetism.    It  is  least  in  very  pure  iron ;  but  if 
the  iron  is  hard,  magnetic  poles  are  slowly  developed,  and  on 
removing  the  inducing  magnet  the  iron  as  slowly  returns  to  its 
former  neutral  state. 

This  property  of  iron  which  obstructs  the  development  of 
magnetism  in  it,  and  which  retards  its  return  to  a  neutral  state, 
is  called  coercive  force.  In  pure  iron,  well  annealed,  the  coercive 
force  is  feeble.  It  appears  in  wrought  iron  not  carefully  prepared ; 
it  is  very  great  in  cast  iron,  and  is  greatest  in  steel  of  the  hardest 
temper. 

It  is,  therefore,  difficult  to  make  a  strong  magnet  of  a  steel 
bar  by  ordinary  induction,  unless  it  is  quite  thin ;  but  after  the 
development  has  once  been  made,  the  bar  becomes  a  permanent 
magnet,  and  may  by  care  be  used  as  such  for  years. 

571.  Change  in  the  Coercive  Force. — The  coercive  force 


362  MAGNETISM. 

is  weakened  by  any  cause  which  excites  a  tremulous  or  vibratory 
motion  among  the  particles  of  the  steel.  This  happens  when  the 
bar  is  struck  by  a  hammer,  so  as  to  produce  a  ringing  sound,  which 
indicates  that  the  particles  are  thrown  into  a  vibratory  motion. 
The  passage  of  an  electric  discharge  through  a  steel  bar  under  the 
influence  of  a  magnet,  overcomes  the  coercive  force  for  the  time 
being,  and  permanent  magnetism  is  developed.  Heat  produces 
the  same  effect ;  and  hence  a  steel  bar  is  conveniently  magnetized 
by  heating  it  to  redness,  placing  it  under  a  powerful  inductive 
influence,  and  then  hardening  it  by  sudden  cooling.  The  coercive 
force  is  thus  neutralized  by  heat,  till  the  development  takes  place, 
when  it  is  restored,  and  the  bar  is  a  permanent  magnet. 

A  magnet,  however,  loses  its  power  by  the  same  means  as, 
during  the  process  of  induction,  were  used  to  develop  it.  Accord- 
ingly, any  mechanical  concussion  or  rough  usage  impairs  or  destroys 
the  power  of  a  magnet.  By  falling  on  a  hard  floor,  or  by  being- 
struck  with  a  hammer,  it  is  injured.  Heat  produces  a  similar 
effect.  A  boiling  heat  weakens,  and  a  red  heat  totally  destroys 
the  magnetism  of  a  needle. 

572.  Magnetism  not  Transferred,  but  only  Developed. 

— This  is  strikingly  proved  by  the  fact  that  if  a  magnet  be  divided, 
even  at  the  neutral  point,  where  there  is  no  sign  of  magnetism, 
the  parts  instantly  become  complete  magnets,  two  unlike  poles 
manifesting  themselves  at  the  place  of  fracture,  whose  strength  is 
equal  to  that  of  the  poles  of  the  original  magnet. 

Both  polarities  seem  to  exist  at  every  point,  each  neutralizing 
any  external  manifestations  of  the  other  (Art.  569) ;  but  when  the 
bar  is  divided  at  any  point  the  two  polarities  at  that  point  are 
free  to  act  upon  magnetic  bodies. 

It  is  not  necessary  that  the  particles  should  be  united  by 
cohesion  into  a  solid  bar.  A  magnet  can  be  formed  by  filling  a 
brass  tube  with  iron  filings  and  sand,  or  by  forming  a  rod  of 
cement  mixed  with  filings,  and  then  subjecting  them  to  inductive 
influence.  Fig.  323  will  give  an  idea  of  the  probable  structure  of 

FIG.  323. 


every  magnet.  Each  particle  of  it  is  a  complete  magnet,  the  like 
poles  of  all  are  turned  the  same  way,  and  unlike  poles  are  there- 
fore contiguous  to  each  other,  and  each  acts  inductively  on  the 
next. 

573.  Magnetic   Intensity.— To  deduce  an  expression  for 


INTENSITY    AND    DISTANCE.  363 

the  mutual  attraction  of  two  poles,  it  is  necessary  to  adopt  some 
unit  of  measure. 

The  British  unit  is  the  force  which,  acting  upon  one  grain  of 
matter  during  one  second,  will  give  to  it  a  velocity  of  one  foot  per 
second.  This  is  found  to  be  about  .03106  grain  (Art.  34). 

The  centimetre  gramme-second  unit,  usually  written  C.  G.  8. 
unit,  is  that  force  which,  acting  during  one  second  upon  a  mass 
of  one  gramme,  imparts  to  it  a  velocity  of  one  centimetre  per 
second.  This  latter  unit  is  called  a  dyne  and  is  nearly  7|T 
gramme.  Adopting  either  of  these  units,  we  define  a  unit  mag- 
netic pole  to  be  a  pole  of  such  strength  that  it  will  attract  another 
equal  pole,  placed  at  a  unit's  distance,  with  the  unit  of  force. 

Suppose  each  of  three  unit  poles,  #,  b,  c  (Fig.  324),  to  be  dis- 
tant one  foot  from  each  of  two  other  unit  poles,  d  and  e.     The 
attraction  between  a  and  d  is  one 
unit,  and  between  a  and  e  one  IG' 

unit  also,  hence  the  attraction  be- 
tween  a  and  d  e,  taken  together, 
is  two  units  ;  in  like  manner  the 
attraction  between  b  and  d  e,  is  two  units,  and  also  between  c  and 
d  e,  it  is  two  units.  Hence  the  total  attraction  between  the  two 
groups  is  3  x  2  =  6  units.  If  now  we  conceive  the  three  poles 
a,  b,  c,  to  be  united  into  one  pole,  it  will  have  a  strength  three 
times  that  of  any  one  of  the  unit  poles ;  and  d  and  e  may  be  united 
into  one  pole  of  strength  twice  that  of  one  pole  alone.  The 
mutual  attraction  will  be  the  same  as  before,  3x2  =  6.  In 
general,  calling  the  strength  of  any  pole  as  compared  with  the 
unit  pole,  m,  and  that  of  any  other  pole  m',  the  mutual  attraction 
will  be  represented  by  their  product,  /  =  m  m'. 

574.  Magnetic  Intensity  and  Distance.— The  law  of  the 

magnetic  force  is  the  following  : 

Tlie  intensity  of  the  magnetic  force,  whether  attraction  or  repul- 
sion, varies  inversely  as  the  square  of  the  distance. 

The  law  in  the  case  of  the  repulsion  of  like  poles  is  readily 
proved  by  Coulomb's  torsion  balance,  which  is  figured  and  de- 
scribed under  Electricity.  The  angle  of  torsion  is  used  as  a 
measure  of  the  repulsion,  and  it  is  found  that  the  wire  must  be 
twisted  through  four  times  as  large  an  angle  to  bring  the  poles  to 
one-half  the  distance,  and  nine  times  as  large  an  angle  to  bring 
them  to  one-third  the  distance,  &c.,  the  force  increasing  as  the 
square  of  the  distance  diminishes. 

To  prove  the  law  for  the  attraction  of  opposite  poles,  the 
vibrations  of  a  needle  are  counted,  when  it  is  placed  at  different 
distances  from  a  magnet.  The  square  of  the  number  made  in  a 


364 


MAGNETISM. 


given  time  is  a  measure  of  the  attractive  force,  just  as  the  square 
of  the  number  of  vibrations  of  a  pendulum  is  a  measure  of  the 
force  of  gravity  (Art.  163). 

In  each  of  these  experiments,  the  magnetic  influence  of  the 
earth  upon  the  needle  must  be  eliminated,  in  order  to  obtain  a, 
correct  result. 

To  determine  the  mutual  attraction  or  repulsion  between  two 
magnets  of  strength  m  and  m',  and  at  a  distance  from  each  other 
of  d  feet,  it  will  only  be  necessary  to  combine  the  results  of  this 
and  the  preceding  Article.  Calling  F  the  attraction  between  the 
two  poles,  under  this  new  condition  we  have 

m  m' 


FIG.  325. 


In  applying  this  formula  any  increase  of  the  strength  of  either 
pole  by  induction  must  be  borne  in  mind  ;  thus,  if  a  magnet  and 
a  piece  of  soft  iron  be  considered,  it  must  be  remembered  that  any 
change  in  the  strength  of  the  magnet  produces  a  like  change  in 
the  induced  magnetism  of  the  iron. 

575.  Equilibrium  of  a  Needle  near  a  Magnet.  —  If  a 

.small  needle,  free  to  revolve,  be  placed  near  the  pole  of  a  magnet, 
so  that  its  centre  is  in  the  axis  of  the  magnet  produced,  it  will 
place  itself  in  the  line  of  that  axis.  For  suppose  that  N  8  (Fig. 

325)  is  a  large  magnetic 
bar,  and  n  s  a  small  needle 
suspended  near  the  north 
pole  of  the  magnet,  with 
its  centre  in  the  axis  of 
the  bar  produced  at  0;  it 
will  be  seen  that  the  ac- 
tion of  the  pole  of  the 
magnet  is  such  as  to  bring 
the  needle  into  a  line  with 

the  magnet.  The  action  of  the  pole  .2^  upon  the  needle  tending 
to  give  it  this  direction  (since  it  repels  n  and  attracts  s),  is 
equal  to  the  sum  of  its  actions  upon  both  poles.  The  pole  S, 
by  repelling  s,  and  attracting  n,  tends  to  reverse  this  posi- 
tion, but  on  account  of  greater  distance,  its  force  is  less  than 
that  of  N. 

If  the  centre  of  the  needle  is  in  a  line  perpendicular  to  the  bar 
at  its  middle  point,  the  needle  will  be  in  equilibrium  when 
parallel  to  the  bar  with  its  poles  in  contrary  order.  Thus  suppos- 
ing the  needle  to  be  suspended  at  &,  it  will  be  seen  that  the  actions 
of  both  poles  of  the  magnet  conspire  to  move  n  to  the  left,  and  s  to 


MAGNETIC    CURVES. 


365 


the  right ;  and  as  these  forces  are  equal,  equilibrium  takes  place 
only  when  the  needle  is  parallel  to  the  bar. 

At  intermediate  points  the  needle  will  assume  all  possible 
inclinations  to  the  axis  of  the  bar,  each  position  being  determined 
by  the  resultant  of  the  four  forces  which  act  on  the  needle.  In 
Fig.  326  are  indicated  some  of  the  positions  which  the  needle  takes 

FIG.  326. 


in  being  carried  round  the  magnet.    While  it  goes  once  round  the 
magnet,  it  makes  two  revolutions  on  its  own  axis. 

It  is  to  be  observed  that  in  all  positions  the  needle  tends,  as  a 
whole,  to  move  toward  the  bar,  since  the  attractions  always 
exceed  the  repulsions. 

576.  Magnetic  Curves. — All  the  foregoing  cases  are  shown 
at  once  by  iron  filings  strewn  on  paper  or  parchment,  which  is 
stretched  on  a  frame  and  placed  near  a  magnet.  Let  the  paper 
be  slightly  jarred,  while  the  magnet  lies  parallel  to  it,  either  above 
or  below,  and  all  the  inclinations  of  the  needle  will  be  represented 
by  the  particles  of  iron  arranged  in  curves  from  pole  to  pole  (Fig. 
327).  Near  the  poles  of  the  magnet  the  filings  stand  up  on  the- 

FIG.  327. 


paper  at  various  inclinations.  These  are  the  extremities  of  still 
other  curves,  which  would  be  formed  in  all  possible  planes  passing 
through  the  axis  of  the  magnet,  provided  the  filings  could  float 


366 


MAGNETISM. 


suspended  in  the  air,  while  the  magnet  is  placed  in  the  midst  of 
them.     These  are  called  magnetic  curves. 

When  the  magnet  is  below  the  paper,  the  particles  move  away 
from  the  area  over  the  poles,  as  in  Fig.  327  ;  but  when  it  is  above, 
they  gather  in  a  cluster  under  each  pole.  This  singular  difference 
arises  from  the  force  of  gravity  acting  on  the  filaments,  which  are 
raised  up  on  the  paper,  and  which  lean,  in  the  former  case,  from 
each  other,  and  in  the  latter,  toward  each  other. 

577.  Lines  of  Force. — Magnetic  Field. — Upon  consider- 
ing the  action  of  one  magnet  upon  others,  as  described  in  Art. 
575,  and  as  graphically  exhibited  in  Fig.  326,  we  discover  that  in 
the  space  surrounding  the  magnet  exists  a  force  which  either 
repels  or  attracts  another  magnet,  and  that  the  centre  of  this 
force  is  at  the  pole  of  the  magnet  which  we  are  considering.  The 
space  within  which  this  force  is  manifested  is  called  the  magnetic 
field,  and  the  intensity  of  the  attraction  or  repulsion  of  the 
magnet  at  any  point  in  this  surrounding  space,  is  called  also  the 
intensity  of  the  field  at  that  point. 

The  curved  and  straight  lines  (Fig.  327)  in  which  the  iron 
filings  arrange  themselves,  are  the  lines  of  force;  that  is  to  say,  the 
lines  along  which  the  force  acts.  If  we  could  conceive  of  a  single 
magnetic  pole,  without  its  opposite  pole,  existing  in  space  not 
influenced  by  other  magnets,  the  lines  of  force  in  that  case  would 
be  straight  lines  radiating  from  such  pole.  The  points  of  equal 
intensity  in  a  magnetic  field  would,  taken  together,  constitute  a 
surface  of  uniform  intensity,  called  an  equipotential  surface, 
which  in  the  case  supposed  would  be  the  surface  of  a  sphere. 
The  expressions  magnetic  field,  lines  of  force,  equipotential  sur- 
face, are  used  in  the  sense 
which  has  been  illustrated, 
however  the  magnetic  field 
may  be  modified  by  other 
fields  brought  near  the  first. 
The  lines  of  force  are  every- 
where perpendicular  to  the 
equipotential  surface. 

Fig.  328  illustrates  the 
lines  of  force  in  the  mag- 
netic field  between  the  un- 
like poles  of  two  magnets, 
the  plane  of  the  paper  being 
a  section  of  the  field  through 
the  common  axis  of  the 
magnets.  Fig.  329  gives  a  similar  section  of  the  field  between  two 


FIG.  328. 


DIAMAGNETISM. 


367 


like  poles  of  equal  intensity,  the  repulsive  forces  being  in  equi- 
librium along  the  line  c  d. 

The  direction  of  a  line  of  force  at  any  given  point  is  the  direc- 
tion in  which  a  single  pole  would  move  if  free  ;  and  the  intensity 
of  the  field  at  any  point  is  proportional  to  the  force  which  it 
exerts  upon  the  free  pole.  In  Art.  573  we  learn  that  the  mutual 
action  of  two  magnets  is  represented  by  m  m' ;  now  call  the 
intensity  of  the  field  at  any  point  M,  and  that  of  the  free  pole  m', 
and  we  have  as  before  f  =  M  m'. 

A  field  of  unit  intensity  is  that  which  acts  with  the  unit  of 
force  upon  the  unit  pole.  The  intensity  of  the  field  is  equal  to 
the  strength  of  the  pole  divided  by  the  square  of  the  distance,  or 

M  =  — ;   hence,  a  unit  field  is  at  a  unit  distance  from  a  unit 

pole. 

If  the  lines  of  force  are  parallel,  equal,  and  equidistant,  the 
field  is  called  a  uniform  field,  and  the  equipotential  surfaces  be- 
come parallel  planes.  A  small  portion  of  a  magnetic  field,  at  a 
great  distance  from  a  pole,  may  be  considered  as  uniform ;  the 
magnetic  field  due  to  terrestrial  magnetism  may  be  considered  as 
uniform  at  any  one  place,  the  lines  of  force  being  indicated  by  the 
position  assumed  by  a  magnetized  needle  perfectly  balanced  upon 
its  centre  of  gravity. 

578.  Position    of   a  Needle   Restrained  by  a    Rigid 
Axis. — Suppose  a  needle  to  be  mounted  upon  an  axis  through 
its  middle  point,  at  right  angles  to  its  length,  so  as  to  be  free  to 
move  only  in  a  plane  perpendicular  to  such  axis.     If  it  be  placed 
with  this  axis  perpendicular  to  the  lines  of  force  of  a  field,  it  will 
place  itself  parallel   to   them 

(Fig.  326) ;  if  the  axis  be 
placed  parallel  to  the  lines 
of  force,  the  needle  will  re- 
main in  any  position,  in  its 
plane  of  motion,  which  we 
give  to  it,  as  in  a  and  b,  Fig. 
330.  If  the  axis  be  inclined 
to  the  lines  of  force,  since  the  needle  will  take  that  position 
which  most  nearly  approaches  parallelism  to  them,  it  must  turn 
till  it  makes  the  least  possible  angle  with  them;  such  position 
in  the  case  represented  in  c,  Fig.  330,  being  evidently  in  the 
plane  of  the  section  of  the  field  represented  by  the  page, 

579.  Diamagnetism. — Thus  far  only  those  substances  which 
-nro  attracted  by  either  pole  of  a  magnet  have  been  called  magnetic  ; 


368 


MAGNETISM. 


FIG.  331. 


but  as  all  substances  seem  to  be  influenced  by  magnetic  force,  a 
new  definition  becomes  necessary. 

Those  substances  which  are  attracted  by  a  magnetic  pole,  or 
which  in  a  magnetic  field  tend  to  move  from  places  of  less  to  places 
of  greater  intensity,  are  called  Paramagnetic. 

TJiose  substances  luhich  are  repelled  by  either  pole  indifferently, 
or  which  move  from  places  of  greater  intensity  to  places  of  less  in- 
tensity in  the  field,  are  called  Diamagnetic.  • 

All  substances  may  be  arranged  in  one  or  the  other  of  these 
two  classes.  A  very  powerful  magnet  is  needed  to  show  the- 
different  effects  produced  upon  paramagnetic  and  diamagnetic 
substances.  Let  N  and  89  Fig.  331,  be  the  opposite  poles  of 
a  large  electro-magnet,  and  let  a  very  small  sphere  of  iron  c  be 

suspended  by  a  thread,  two  or 
three  feet  long,  so  as  to  rest  at  c, 
midway  between  the  poles,  but 
not  in  the  axial  line  N  8.  Upon 
making  N  and  8  magnets,  the 
sphere  will  be  drawn  into  the  line 
N  S,  and  will  swing  towards 
whichever  pole  happens  to  be 
nearest.  Remove  the  iron  sphere  c, 
and  replace  it  by  one  of  bismuth ; 
now  upon  again  magnetizing  the 

poles  N  and  8,  the  sphere  c  will  be  repelled  from  the  axial  line 
N  8  a  perceptible  distance. 

Eeplace  the  sphere  by  an  iron  needle  suspended  with  its  centre 
in  the  axial  line  N  8.  It  will  place  itself  with  its  longest  dimen- 
sion in  the  line  N  8,  or  axially.  Substitute  for  the  iron  needle 
one  of  bismuth,  or  phosphorus,  and  the  new  needle  will  place 
itself  with  its  length  at  right  angles  to  the  line  N  8,  or  equatori- 
ally.  The  iron  needle,  under  the  attraction  of  the  poles,  approaches 
as  near  them  as  possible,  and  to  do  this  must  set  axially ;  while 
the  bismuth  needle  being  repelled  must  set  equatorially  in  order 
to  recede  as  far  as  possible  from  the  poles. 

Among  diamagnetic  substances  bismuth,  phosphorus,  and 
antimony  are  the  most  marked  ;  but  the  force  of  repulsion,  in  the 
case  of  bismuth  even,  is  only  a  very  small  fraction  of  the  force  of 
attraction  of  iron  under  like  conditions,  being  less  than  gooiooo 
of  the  latter,  according  to  Weber. 

580.  Influence  of  the  Surrounding  Medium.— If  a  para- 
magnetic needle  be  suspended  in  a  fluid  more  strongly  paramag- 
netic than  itself,  it  will  seem  to  be  diamagnetic;  and  if  a  dia- 
magnetic needle  be  suspended  in  a  fluid  more  diamagnetic  than 
itself,  it  will  seem  to  be  paramagnetic. 


DIAMAGNETIC    POLARITY.  360 

The  air  is  less  strongly  paramagnetic  than  a  weak  solution  of 
iron,  and  hence  when  a  tube  containing  such  solution  is  properly 
suspended  between  the  poles  of  a  magnet  it  will  arrange  itself 
axially ;  but  if  the  same  tube  be  now  suspended  in  a  stronger 
solution  of  iron  it  will  set  equator! ally. 

581.  Diamagnetic  Polarity. — Faraday's  experiments  to  de- 
tect diamagnetic  polarity,  owing  to  lack  of  delicacy  in  his  instru- 
ments, gave  no  evidence  of  its  existence.     Other  physicists  also 
failed  to  detect  any  manifestations  of  polarity,  till  Tyndall  in  1855 
established  the  fact  of  diamagnetic  polarity  by  series  of  experi- 
ments, essentially  as  follows : 

Let  JVand  S  (Fig.  332)  be  the  unlike  poles  of  two  electro- 
magnets, and  H  a  helix,  or  coil  of  wire,  of 
internal  diameter  sufficient  to  allow  some  FlG- 

vibration  of  a  needle  a  a'  suspended  within 
it.  Now  if  a  a'  be  a  needle  of  iron  it  will 
become  a  magnet  when  a  current  of  elec- 
tricity flows  through  the  wire,  as  will  be 
explained  under  Dynamic  Electricity. 
Suppose  a  to  be  a  marked  pole,  and  a'  an 
unmarked  pole,  under  the  given  condi- 
tions ;  now  if  the  flow  of  the  current  be  reversed  a  becomes  an 
unmarked  pole  and  a'  a  marked  pole,  the  poles  being  reversed  by 
the  reversal  of  the  current,  as  shown  by  the  corresponding  attrac- 
tions and  repulsions  between  N  and  «,  or  8  and  a'.  When  a 
needle  of  bismuth  is  substituted  for  the  iron,  analogous  but 
directly  opposite  effects  are  produced  ;  that  direction  of  the  cur- 
rent which  in  iron  produced  attraction,  now  causes  repulsion  in 
the  case  of  bismuth,  and  that  current  which  produced  repul- 
sion now  produces  attraction.  Later  experiments  with  a  different 
arrangement  of  helices  and  needles  gave  much  more  striking 
results.  Liquids  were  also  experimented  upon,  inclosed  in  thin 
glass  tubes.  Water  was  found  to  exhibit  diamagnetic  polarity. 

582.  Axis  in  Line  of  Greatest  Density. — If  the  body 
experimented  upon  is  not  homogeneous,  then  the  magnetic  axis 
coincides  with  the  line  of  greatest  density. 

In  crystalline  substances  the  axis  is  parallel  to  the  planes  of 
cleavage,  so  that  a  paramagnetic  body  will  assume  a  position  with 
its  planes  of  cleavage  axial,  while  a  diamagnetic  substance  will 
place  its  planes  of  cleavage  equatorially. 

If  by  pressure  the  density  of  a  substance  be  made  greatest  in  a 
line  at  right  angles  to  the  planes  of  cleavage  this  line  will  deter- 
mine the  set  of  the  body ;  and  by  repeated  pressures  in  various 
24 


370  MAGNETISM. 

directions  successive  changes  in  the  direction  of  the  axis  may  be 
produced. 

In  such  experiments  the  most  striking  effects  are  secured  when 
the  three  dimensions  of  the  body  do  not  differ  greatly  from  each 
other. 

583.  Molecular  Changes. — A  soft  iron  bar  may  be  made  a 
powerful  magnet  by  causing  a  current  of  electricity  to  circulate 
around  it,  as  will  be  more  fully  explained  under  Electro  Mag- 
netism. 

At  the  instant  of  completing  the  circuit  of  the  electric  current 
around  the  bar,  a  sharp  click  is  audible,  and  at  the  instant  when 
the  circuit  is  broken  the  sound  is  heard  again. 

If  a  bar  be  carefully  measured  before  making  the  circuit,  and 
again  during  the  passage  of  the  current  around  it,  it  will  be 
found  that  magnetization  has  caused  an  increase  of  length  and  a 
decrease  of  cross-section  of  the  bar,  the  ratio  of  these  changes 
being  such  that  the  volume  of  the  bar  has  been  unaltered.  These 
dimensional  changes  are  very  slight,  and  can  be  shown  only  by 
very  delicate  experiments. 

To  account  for  these  effects,  we  may  consider  the  bar  to  be 
made  up  of  separate  particles,  as  in  Art.  572  ;  now  when  the  bai- 
ls magnetized  these  particles  tend  to  arrange  themselves  with 
their  longest  dimensions  parallel  to  the  magnetic  axis.  Instead 
of  the  separate  particles  of  Art.  572  substitute  the  infinitely 
minute  crystals  of  the  iron  bar,  with  a  tendency  to  place  the 
longest  axes  parallel  to  the  length  of  the  bar,  and  we  have  De  La 
Hive's  explanation  of  the  phenomena. 


CHAPTER    II. 

RELATIONS  OF  THE  MAGNET  TO  THE  EARTH. 

584.  Declination  of  the  Needle.— When  the  needle  is 
balanced  horizontally,  and  free  to  revolve,  it  does  not  generally 
point  exactly  north  and  south  ;  and  the  angle  by  which  it  deviates 
from  the  meridian  is  called  the  declination.  A  vertical  circle 
coincident  with  the  direction  of  the  needle  at  any  place  is  called 
the  magnetic  meridian.  As  the  angle  between  the  magnetic  and 
the  geographical  meridians  is  generally  different  for  different 
places,  and  also  varies  at  different  times  in  the  same  place,  the 


ISOGONIC    CURVES. 


371 


word  variation  expresses  these  changes  in  declination,  though  it  is 
much  used  as  synonymous  with  declination  itself. 

The  force  which  causes  the  needle  to  set  in  the  magnetic 
meridian  is  merely  directive. 

If  the  needle  be  weighed  before  it  is  magnetized  and  again 
after  it  has  been  made  a  magnet,  no  change  of  weight  can  be 
detected,  proving  that  the  earth's  attraction  for  one  pole  is  exactly 
equal  to  its  repulsion  of  the  other.  This  may  also  be  shown  by 
attaching  a  magnet  to  a  cork  and  thus  floating  it  upon  water.  It 
will  set  in  the  magnetic  meridian  but  will  show  no  tendency  to 
move  across  the  water  towards  the  north,  nor  in  any  other  direc- 
tion. This  effect  is  due  to  the  earth's  uniform  magnetic  field. 
The  magnetic  pole  of  the  earth  being  practically  at  an  infinite 
distance,  the  forces  of  attraction  and  repulsion,  being  equal,  con- 
stitute a  couple  (Art.  56). 

585.  Isogonic  Curves. — This  name  is  given  to  a  system  of 
lines  imagined  to  be  drawn  through  all  the  points  of  equal  decli- 
nation on  the  earth's  surface.  We  naturally  take  as  the  standard 
line  of  the  system  that  which  connects  the  points  of  no  declina- 
tion, or  the  isogonic  of  0°  (Fig.  333).  Commencing  at  the  north 

FIG.  333. 


pole  of  dip,  about  Lat.  70°,  Lon.  96°,  it  runs  in  a  general  direc- 
tion E.  of  S.,  through  Hudson's  Bay,  across  Lake  Erie,  and  the 
State  of  Pennsylvania,  and  enters  the  Atlantic  Ocean  on  the  coast 
of  North  Carolina.  Thence  it  passes  east  of  the  West  India 
Islands,  and  across  the  N.  E.  part  of  South  America,  pursuing  its 
course  to  the  south  polar  regions.  It  reappears  in  the  eastern 
hemisphere,  crosses  Western  Australia,  and  bears  rapidly  westward 
across  the  Indian  Ocean,  and  then  pursues  a  northerly  course 
across  the  Caspian  Sea  to  the  Arctic  Ocean.  There  is  also  a 


MAGNETISM. 

detached  line  of  no  declination,  lying  in  eastern  Asia  and  the 
Pacific  Ocean,  returning  into  itself,  and  inclosing  an  oval  area  of 
40°  N.  and  S.  by  30°  E.  and  W.  Between  the  two  main  lines  of 
no  declination  in  the  Atlantic  hemisphere,  the  declination  is  west- 
war  dy  marked  by  continued  lines,  in  Fig.  333  ;  in  the  Pacific 
hemisphere,  outside  of  the  oval  line  just  described,  it  is  eastward, 
marked  by  dotted  lines.  Hence,  on  the  American  continent,  in 
all  places  east  of  the  isogonic  of  0°,  the  marked  pole  of  the  needle 
declines  westward,  and  in  all  places  west  of  it,  the  marked  pole 
declines  eastward;  on  the  other  continent  this  is  reversed,  as 
shown  by  the  figure. 

Among  other  irregularities  in  the  isogonic  system,  there  are 
two  instances  in  which  a  curve  makes  a  wide  sweep,  and  then 
intersects  its  own  path,  while  those  within  the  loop  thus  formed 
return  into  themselves.  One  of  these  is  the  isogonic  of  8°  40'  E., 
which  intersects  in  the  Pacific  Ocean  west  of  Central  America ; 
the  other  is  that  of  22°  13'  W.,  intersecting  in  Africa. 

In  the  northeastern  part  of  the  United  States  the  declination 
has  long  been  a  few  degrees  to  the  west,  with  very  slow  and 
somewhat  irregular  variations. 

586.  Secular  and  Annual  Variation.  —  The  declination 
of  the  needle  at  a  given  place  is  not  constant,  but  is  subject  to  a 
slow  change,  which  carries  it  to  a  certain  limit  on  one  side  of  the 
meridian,  when  it  becomes  stationary  for  a  time,  and  then  returns, 
and  proceeds  to  a  certain  limit  on  the  other  side  of  it,  occupying 
two  or  three  centuries  in  each  vibration.     At  London,  in  1580,  the 
declination  was  11^°  E.  ;  in  1657,  it  was  0°;  after  which  time  the 
needle  continued  its  western  movement  till  1818,  when  the  decli- 
nation was  24£°  W. ;  since  then  the  needle  has  been  moving  slowly 
eastward,  and  in  1879,  at  Kew,  the  declination  was  19°  7'  west. 

The  entire  secular  vibration  will  probably  last  more  than  three 
centuries.  The  average  variation  from  1580  to  1818  was  9'  10" 
annually.  But,  like  other  vibrations,  the  motion  is  slowest  to- 
ward the  extremes. 

There  has  also  been  detected  a  small  annual  variation,  in  which 
the  needle  turns  its  marked  pole  a  few  minutes  to  the  east  of  its 
mean  position  between  April  and  July,  and  to  the  west  the  rest 
of  the  year.  This  annual  oscillation  does  not  exceed  15  or  18 
minutes. 

587.  Diurnal  Variation. — The  needle  is  also  subject  to  a 
small  daily  oscillation.     In  the  morning  the  marked  end  of  the 
needle  has  a  variation  to  the  east  of  its  mean  position  greater  than 
at  any  other  part  of  the  day.     During  winter  this  extreme  point 
is  attained  at  about  8  o'clock,  but  as  early  as  7  o'clock  in  the  sum- 


ISOCLINIC    CURVES. 


373 


FIG.  334. 


mer.  After  reaching  this  limit  it  gradually  moves  to  the  west, 
and  attains  its  extreme  position  about  3  o'clock  in  winter,  and 
1  o'clock  in  summer.  From  this  time  the  needle  again  returns 
eastward,  reaching  its  first  position  about  10  P.  M.,  and  is  almost 
stationary  during  the  night.  The  whole  amount  of  the  diurnal 
variation  rarely  exceeds  12  minutes,  and  is  commonly  much  less 
than  that.  These  diurnal  changes  of  declination  are  connected 
with  changes  of  temperature,  being  much  greater  in  summer  than 
in  winter.  Thus,  in  England  the  mean  diurnal  variation  from 
May  to  October  is  10  or  12  minutes,  and  from  November  to 
April,  only  5  or  6  minutes. 

588.  Dip  of  the   Needle.— A  needle  first  balanced  on  a 
horizontal  axis,  and  then  magnetized  and  placed  in  the  magnetic 
meridian,  assumes  a  fixed  relation  to  the  horizon,  one  pole  or  the 
other  being  usually  depressed  below  it. 

The  axis  of  the  needle  must  be  placed  very  accurately  at  right 
angles  to  the  plane  of  the 
magnetic  meridian,  or  false 
indications  will  be  given ; 
if  the  axis  of  suspension 
were  placed  in  the  plane 
of  the  meridian  the  angle 
of  depression  would  be  90° 
at  all  places  on  the  earth's 
surface  (Art.  578). 

The  angle  of  depres- 
sion is  called  the  dip  of  the 
needle.  Fig.  334  represents 
the  dipping  needle,  with  its 
adjusting  screws  and  spirit- 
level  ;  and  the  depression 
may  be  read  on  the  grad- 
uated scale.  After  the  hori- 
zontal circle  m  is  leveled  by 
the  foot-screws,  the  frame 
A  is  turned  horizontally  till 
the  vertical  circle  M  is  in 

the  magnetic  meridian.  For  north  latitudes,  the  marked  end  of 
the  needle  is  depressed,  as  a  in  the  figure. 

589.  Isoclinic  Curves. — A  line  passing  through  all  points 
where  the  dip  of  the  needle  is  nothing,  i.  e.,  where  the  dipping 
needle  is  horizontal,  is  called  the  magnetic  equator  of  the  earth. 
It  can  be  traced  in  Fig.  335  as  an  irregular  curve  around  the 


374  .         MAGNETISM. 

earth  in  the  region  of  the  equator,  nowhere  departing  from  it 
more  than  about  15°.     At  every  place  north  of  the  magnetic 

FIG.  335. 


equator  the  marked  pole  of  the  needle  descends,  and  south  of  it 
the  unmarked  pole  descends ;  and,  in  general,  the  greater  the  dis- 
tance, the  greater  is  the  dip.  Imagine  now  a  system  of  lines, 
each  passing  through  all  the  points  of  equal  dip ;  these  will  be 
nearly  parallel  to  the  magnetic  equator,  which  may  be  regarded 
as  the  standard  among  them.  These  magnetic  parallels  are  called 
the  isoclinic  curves  ;  they  somewhat  resemble  parallels  of  latitude, 
but  are  inclined  to  them,  conforming  to  the  oblique  position  of 
the  magnetic  equator.  In  the  figure,  the  broken  lines  show  the 
dip  of  the  south  pole  of  the  needle ;  the  others,  that  of  the  north 
pole.  The  points  of  greatest  dip,  or  dip  of  90°,  are  called  the  poles 
of  dip.  There  is  one  in  the  northern  hemisphere,  and  one  in  the 
southern.  The  north  pole  .of  dip  was  found,  by  Capt.  James  C. 
Ross,  in  1831,  to  be  at  or  very  near  the  point,  70°  14'  K  ;  96°  40' 
W.,  marked  x  in  the  figure.  The  south  pole  is  not  yet  so  well 
determined. 

At  the  poles  of  dip  the  horizontal  needle  loses  all  its  directive 
power,  because  the  earth's  magnetism  tends  to  place  it  in  a  verti- 
cal line,  and,  therefore,  no  component  of  the  force  can  operate  in 
a  horizontal  plane.  The  isogonic  lines  in  general  converge  to  the 
two  dip-poles  ;  but,  for  the  reason  just  given,  they  cannot  be  traced 
quite  to  them. 

The  dip  of  the  needle,  like  the  declination,  undergoes  a  varia- 
tion, though  by  no  means  to  so  great  an  extent. 

In  1576,  the  date  of  its  discovery,  the  dip  at  London  was 
71°  50' ;  it  increased  to  a  maximum  of  74°  42'  in  1723,  since 
which  time  it  has  gradually  decreased.  In  1879  the  dip  at  Kew 
was  67°  42'. 


1SODYNAMIC    CURVES.  375 

In  the  course  of  250  years,  it  has  diminished  about  five  degrees 
in  London.  In  1820  it  was  about  70°,  and  diminishes  from  two 
to  three  minutes  annually. 

Since  the  dip  at  a  given  place  is  changing,  it  cannot  be  supposed 
thd't  the  poles  are  fixed  points ;  they,  and  with  them  the  entire 
system  of  isoclinic  curves,  must  be  slowly  shifting  their  locality. 

590.  Magnetic  Intensity  of  the  Earth. — The  force  ex- 
erted by  the  magnetism  of  the  earth  varies  in  different  places, 
being  generally  least  in  the  region  of  the  equator,  and  greatest 
in  the  polar  regions.    The  ratio  of  intensity  in  different  places 
is  measured  by  the  number  of  vibrations  which  the  needle  makes 
in  a  given  time.    In  the  discussion  of  the  pendulum,  it  was  proved 
(Art.  163)  that  gravity  varies  as  the  square  of  the  number  of 
vibrations.     For   the   same  reason    the  magnetic  force  at  any 
place  varies  as  the  square  of  the  number  of  vibrations  of  the 
needle  at  that  place,  provided   the  axis  of  suspension   of   the 
needle  be  perpendicular  to  the  lines  of   force.      As  it  is  not 
convenient  to  use   the  dipping  needle   in  this   determination, 
the  oscillations  of  a  horizontal  needle  are  used  instead,  and  the 
intensity  is  computed  from  these.    Thus,  let  /  (Fig.  336)  repre- 
sent the  direction  and  intensity  of  the  earth's  mag- 

"C*Tn.     QQfi 

netic  attraction  at  any  place,  and  d  the  angle  of 
dip.  The  horizontal  component  of  the  inten- 
sity will  be  /  x  cos  d,  and  it  is  this  component 
which  varies  as  the  square  of  the  number  of  oscilla- 
tions of  a  horizontal  needle.  Calling  n  and  n'  the 
number  of  oscillations  per  second  at  two  different 
places,  d  and  d'  the  dip,  and  /  and  /'  the  relative  intensities,  we 
have 

/  cos  d  :  I'  cos  d'  : :  n*  :  n\ 
from  which  is  deduced 

/'  _  cos  d       nP 
I  ' ~  cos  d'       n2' 

591.  Isodynamic   Curves. — After  ascertaining,  by  actual 
observation,  the  intensity  of  the  magnetic  force  in  different  parts 
of  the  earth,  lines  are  supposed  to  be  drawn  through  all  those 
points  in  which  the  force  is  the  same  ;  these  lines  are  called  isody- 
namic  curves,  represented  in  Fig.  337.   These  also  slightly  resemble 
parallels  of  latitude,  but  are  more  irregular  than  the  isoclinic  lines. 
There  is  no  one  standard  equator  of  minimum  intensity,  but  there 
are  two  very  irregular  lines  surrounding  the  earth  in  the  equatorial 
region,  in  some  places  almost  meeting  each  other,  and  in  others 
spreading  apart  more  than  two  thousand  miles,    on   which  the 


376  MAGNETISM. 

magnetic  intensity  is  the  same.  These  two  are  taken  as  the 
standard  of  comparison,  because  they  are  the  lowest  which  extend 
entirely  round  the  globe.  The  intensity  on  them  is  therefore 

FIG.  337. 


called  unity,  marked  1  in  the  figure.  In  the  wide  parts  of  the 
belt  which  they  include — lying  one  in  the  southern  Atlantic,  and 
the  other  in  the  northern  Pacific  oceans — there  are  lines  of  lower 
intensity  which  return  into  themselves,  without  encompassing  the 
earth.  In  approaching  the  polar  regions,  both  north  and  south, 
the  curves,  retaining  somewhat  the  form  of  the  unit  lines,  are 
indented  like  an  hour-glass,  as  those  marked  1.7  in  the  figure, 
and  at  length  the  indentations  meet,  forming  an  irregular  figure 
8 ;  and  at  still  higher  latitudes,  are  separated  into  two  systems, 
closing  up  around  two  poles  of  maximum  intensity.  Thus  there 
are  on  the  eartli  four  poles  of  maximum  intensity,  two  in  the 
northern  hemisphere  and  two  in  the  southern.  The  American 
north  pole  of  intensity  is  situated  on  the  north  shore  of  Lake 
Superior.  The  one  on  the  eastern  continent  is  in  northern 
Siberia.  The  ratio  of  the  least  to  the  greatest  intensity  on  the 
earth  is  about  as  0.7  to  1.9 ;  that  is,  as  1  to  2%.  In  the  figure, 
intensities  less  than  1  are  marked  by  dotted  lines. 

592.  Magnetic  Charts. — These  are  maps  of  a  country,  or 
of  the  world,  on  which  are  laid  down  the  systems  of  curves  which 
have  been  described.  But  for  the  use  of  the  navigator,  only  the 
isogonic  lines,  or  lines  of  equal  declination,  are  essential.  There 
are  large  portions  of  the  globe  which  have  as  yet  been  too  imper- 
fectly examined  for  the  several  systems  of  curves  to  be  accurately 
mapped.  It  must  be  remembered,  too,  that  the  earth  is  slowly 
but  constantly  undergoing  magnetic  changes,  by  which,  at  any 
given  place,  the  declination,  dip,  and  intensity  are  all  essentially 


AURORA    BOREALIS.  377 

altered  after  the  lapse  of  years.  A  chart,  therefore,  which  would 
be  accurate  for  the  middle  of  the  nineteenth  century,  will  be,  to 
some  extent,  incorrect  at  its  close. 

593.  Magnetic  Observatories. — In  accordance  with  a  sug- 
gestion of  Humboldt,  in  1836,  systematic  observations  have  been 
since  made  upon  terrestrial  magnetism,  in  various  parts  of  the 
world,  in  order  to  deduce  from  them  the  laws  of  its  changes. 
Buildings  have  been  erected  without  any  iron  in  their  construc- 
tion, to  serve  as  magnetic  observatories;   and  the  most  delicate 
magnetometers  have  been  devised  and  used  for  detecting  minute 
oscillations  both  in  the  horizontal  and  vertical  planes,    By  these 
means  has  been  discovered  a  class  of  phenomena  called  magnetic 
storms,  in  which  the  needle  suffers  numerous  and  rapid  disturb- 
ances, sometimes  to  the  extent  of  several  degrees;   and  it  is  a 
remarkable  and  interesting  fact  that  these  disturbances  occur  at 
the  same  absolute  time  in  every  part  of  the  earth. 

594.  Aurora  Borealis. — This  phenomenon  is  usually  accom- 
panied by  a  disturbance  of  the  needle,  thus  affording  visible  indi- 
cations of  a  magnetic  storm;  but  the  contrary  is  by  no  means 
generally  true,  that  a  magnetic  storm  is  accompanied  by  auroral 
light.     The  connection  of  the  aurora  borealis  with  magnetism  is 
manifested  not  only  by  the  disturbance  of  the  needle,  but  also  by 
the  fact  that  the  streamers  are  parallel  to  the  dipping  needle,  as 
is  proved  by  their  apparent  convergence  to  that  point  of  the  sky 
to  which  the  dipping-needle  is  directed.     This  convergence  is  the 
effect  of  perspective,  the  lines  being  in  fact  straight  and  parallel. 

595.  Source  of  the  Earth's  Magnetism.— If  a  needle 
is  carried  round  the  earth  from  north  to  south,  it  takes  approxi- 
mately all  the  positions  in  relation  to  the  earth's  axis  which  it 
assumes  in  relation  to  a  magnetic  bar,  when  carried  round  it  from 
end  to  end  (Art.  575).     At  the  equator  it  is  nearly  parallel  to 
the  axis,  and  it  inclines  at  larger  and  larger  angles  as  the  distance 
from  the  equator  increases ;  and  in  the  region  of  the  poles,  it  is 
nearly  in  the  direction  of  the  axis.    The  earth  itself,  therefore, 
may  be  considered  a  magnet,  since  it  affects  a  needle  as  a  magnet 
does,  and  also  induces  the  magnetic  state  on  iron.     But  it  is 
necessary,  on  account  of  the  attraction  of  opposite  poles,  to  con- 
sider the  northern  part  of  the  earth  as  being  like  the  south  pole 
of  a  needle,  and  the  southern  part  like  the  north  pole.     To  avoid 
this,  the  words  boreal  and  austral  are  applied  to  the  two  magnetic 
states,  and  the  boreal  magnetism  is  the  name  given  to  that  develop- 
ment found  in  the  northern  hemisphere,  and  the  austral  magnetism 
to  that  in  the  southern.     Hence,  it  becomes  necessary,  in  using 


378  MAGNETISM. 

these  names  for  a  magnet,  to  reverse  their  order,  and  to  speak  of 
its  north  pole  as  exhibiting  the  austral,  and  its  south  pole  the 
boreal  magnetism. 

Modern  discoveries  in  electro-magnetism  and  thermo-electricity 
furnish  a  clew  to  the  hypothesis  which  generally  prevails  at  this 
day.  Attention  has  been  drawn  to  the  remarkable  agreement 
between  the  isothermal  and  the  isomagnetic  lines  of  the  globe. 
The  former  descend  in  crossing  the  Atlantic  Ocean  toward 
America,  and  there  are  two  poles  of  maximum  cold  in  the 
northern  hemisphere.  The  isoclinic  and  the  isodynamic  curves 
also  descend  to  lower  latitudes  in  crossing  the  Atlantic  west- 
ward ;  so  that,  at  a  given  latitude,  the  degree  of  cold,  the  mag- 
netic dip,  and  the  magnetic  intensity,  is  each  considerably  greater 
on  the  American  than  on  the  European  coast.  This  is  only  an 
instance  of  the  general  correspondence  between  these  different 
systems  of  curves.  It  has  likewise  been  noticed  (Art.  587)  that 
the  needle  has  a  movement  diurnally,  varying  westward  during 
the  middle  of  the  day,  and  eastward  at  evening,  and  that  this 
oscillation  is  generally  much  greater  in  the  hot  season  than  the 
cold.  It  is  obvious,  therefore,  that  the  development  of  magnetism 
in  the  earth  is  intimately  connected  with  the  temperature  of  its 
surface.  Hence  it  has  been  supposed  that  the  heat  received  from 
the  sun  excites  electric  currents  in  the  materials  of  the  earth's 
surface,  and  these  give  rise  to  the  magnetic  phenomena. 

The  actual  phenomena,  while  presenting  the  analogies  given 
above,  also  indicate  peculiar  conditions  such  that  no  simple  law  of 
magnetic  distribution  can  be  enunciated. 


CHAPTER    III. 

MAGNETIZATION. 

596.  Formation  of  Permanent  Magnets.— Needles  and 
small  bars  may  be  more  or  less  magnetized  by  the  following 
methods,  the  reasons  for  which  will  be  readily  understood  : 

1.  A  feeble  magnetism  may  be  developed  in  a  steel  bar,  by 
causing  it  to  ring  while  held  vertically.  The  earth's  influence 
upon  it,  however,  is  stronger  if  it  is  held,  not  precisely  vertical, 
but  leaning  in  a  direction  parallel  to  the  dipping  needle.  The 
inductive  influence  of  the  earth  explains  the  fact  often  noticed, 
that  rods  of  iron  or  steel  that  have  stood  for  many  years  in  a 
position  nearly  vertical,  as,  for  instance,  lightning-rods,  iron 


THE    HORSESHOE    MAGNET. 


379 


pillars,    stoves,    &c.,    are  found  somewhat  magnetic,    with  the 
marked  pole  downward. 

2.  A  needle   may  be   magnetized  by   simply  suffering  it  to 
remain  in  contact  with  the  pole  of  a  strong  magnet,  or  better, 
in  contact  with  the  opposite  poles  of  two  magnets. 

3.  Place  two  opposite  poles  of  equally  strong  magnets  MM', 
(Pig.  338),  upon  the  middle  of  the  bar  to  be  magnetized  a  b,  and 
draw  them  from  the  middle  to  the  ends.      Return  the  magnets 
MM'  to  the  middle 

of  the  bar  again,  car-  Fia-  338< 

rying  them  around  at 

a  distance  from  the 

bar,  and  repeat  the 

first  stroke.     Do  this 

several    times    upon 

each  face  of  a  I.     The  effects  will  be  intensified  if  a  ~b  be  placed 

upon  two  other  magnets,  as  in  Fig.  338.      This  is  called  the 

method  of  separate  touch. 

4.  If,  instead  of  moving  the  magnets  from  the  middle  of  the 
needle  towards  each  end  simultaneously,  they  be  moved  together, 
first  towards  a  (Fig.  338)  and  then  back  again  to  b,  and  so  on, 
ending  the  strokes  at  the  middle,  the  magnetization  will  be  very 
strong.     This  is  the  method  of  double  touch. 

Either  of  these  two  methods  may  be  applied  to  several  bars  at 
once,  or  to  two  horseshoe  magnets.  The  variations 
in  the  mode  of  application  are  numerous. 

In  order  to  take  advantage  of  the  earth's  induc- 
tive influence,  along  with  that  of  steel  magnets, 
place  the  needle  parallel  to  the  dipping  needle,  and 
draw  the  unmarked  pole  of  one  magnet  over  the 
lower  half,  and  the  marked  pole  of  another  over  the 
upper  half,  with  repeated  and  simultaneous  move- 
ments. 

If  the  steel  bar  be  not  homogeneous,  or  if  any  of 
the  processes  of  magnetization  be  not  carefully  carried 
on,  the  bar  may  have  one  or  more  poles  between  the 
extreme  poles.  These  are  called  consequent  points  or 
poles,  and  are  more  likely  to  be  developed  in  very 
long  bars  than  in  short  ones. 

None  of  these  methods,  however,  are  of  great  prac- 
tical value  at  the  present  day,  since  the  galvanic  cir- 
cuit affords  a  far  readier  and  more  efficient  means  of 
magnetizing  bars. 

The    horseshoe  magnet,    sometimes    called    the 
U-magnet  (Fig.  339),  is  for  many  purposes  a  very  convenient  form, 


PIG.  339. 


380 


MAGNETISM. 


and  originated  in  the  practice  of  arming  the  lodestone  ;  that  is, 
furnishing  it  with  two  pieces  of  soft  iron,  which  are  confined  by 
brass  straps  to  the  poles  of  the  stone,  and  project  below  it,  so  that 
a  bar  and  weight  may  be  attached.  When  a  magnet  has  this 
form,  both  poles  may  be  applied  to  a  body  at  once.  The  £7-mag- 
net,  A  N  S9  being  suspended,  and  the  keeper,  B,  made  of  soft 
iron,  being  attached  to  the  poles,  weights  may  be  hung  upon  the 
hook  C,  to  show  the  strength. 

597.  Compound  Magnets. — Thin  steel  plates  can  be  most 
readily  magnetized.  Many  of  these  bound  together  by  suitable 
clamps  of  a  non -magnetic  substance,  with  their  like  poles  side  by 
side,  constitute  a  compound  magnet.  Since  the  mutual  action  of 
the  like  poles  in  juxtaposition  tends  to  weaken  them,  the  strength 
of  the  compound  magnet  will  never  equal  the  sum  of  the  strengths 
of  its  component  parts,  but  it  will  far  exceed  that  of  any  one  of 
them,  and  also  that  of  a  solid  bar  of  a  mass  equal  to  the  sum  of 
their  masses.  The  best  effects  are  produced  by  using  an  odd 
number  of  plates,  of  lengths  such  that  the  successive  pairs  may 
be  shorter  than  the  preceding,  giving  tapered  poles  as  in  Fig. 

340.  This  rounded  or 
tapered  form  of  pole 
is  also  best  for  single 
bar  and  horseshoe 
magnets. 

598.  Armatures. 
—The  keeper  B  (Fig. 
339),  is  called  an  ar- 
mature. If  two  bar- 
magnets  (Fig.  341)  be 
placed  parallel,  and 

their  poles  be  joined  by  soft  iron  bars  A  and  B,  these  are  called 
armatures  also. 

The  armature  of  a  magnet,  when  in  contact  with  the  poles, 
tends  to  diminish  the  intensity  of  the  surrounding  field.  This 
may  be  readily  shown  by  bringing  a  small  horseshoe  magnet  near 
a  declination  needle  ;  on  removing  the  armature,  the  deviation  of 
the  needle  increases,  and  on  replacing  the  armature  the  needle 
returns  to  its  former  position.  An  electro-magnetic  ring  pro- 
duces no  sensible  field  in  its  neighborhood. 

599.  Preservation  of  Magnets. — That  magnets  may  be 
preserved  in  good  condition  care  should  be  exercised  in  handling 
them.  They  should  never  be  subjected  to  blows,  nor  to  atiy  jar- 
ring action.  They  should  not  be  greatly  heated.  Bar-magnets 


FIG.  341. 


POWER    OF    MAGNETS.  381 

should  always  be  placed  in  the  plane  of  the  magnetic  meridian, 
the  marked  poles  north,  when  laid  away  after  use.  Like  poles  of 
magnets  of  unequal  strength  should  not  be  brought  into  contact, 
lest  the  stronger  should  reverse  the  polarity  of  the  weaker. 
Horseshoe  magnets,  and  bar-magnets  in  pairs,  should  always  have 
their  armatures  in  contact,  when  laid  away.  The  armature  should 
not  be  jerked  away,  but  should  be  removed  by  sliding  off. 

600.  Saturation  of  Magnets. — Up  to  a  certain  point  the 
magnetism  induced  in  a  steel  bar  increases  with  the  strength  of 
the  magnets  used  and  with  the  increase  of  the  number  of  touches  ; 
but  soon  a  limit  is  reached,  and  then  the  bar  is  said  to  be  mag- 
netized to  saturation.     It  is  possible  to  give  a  temporary  increase 
to  the  intensity  of  a  permanent  magnet,  but  when  the  inducing 
cause  is  removed  the  intensity  falls  again,  rapidly  at  first,  and  then 
more  slowly  for  days  and  even  weeks.     If  a  horseshoe  magnet  ba 
suspended,  and  weights  be  hung  upon  the  armature  (Fig.  339),  it 
will  be  found  that  after  a  time  still  other  weights  may  be  added, 
till  finally  a  much  larger  load  will  be  sustained  than  the  magnet 
will  ordinarily  "pick  up."     When  this  loaded  armature  is  torn 
away  the  strength  of  the  magnet  decreases  to  its  normal  con- 
dition. 

601.  Power  of  Magnets. — The  strength  of  a  saturated  horse- 
shoe magnet  is  given  by  the  following  formula  from  Haecker, 
P  =  atyp2,  in  which  P  is  the  maximum  weight  which  the  mag- 
net can   lift,  p  the  weight  of  the  magnet  itself,   and  a  a  con- 
stant depending  upon  the  quality  of  the  steel  and  the  mode  of 
magnetization.     By  means  of  this  formula  we  derive  the  relative 
powers  of  two  magnets,  subject  to  the  same  constant  a.     Suppose 
one  magnet  to  be  eight  times  as  heavy  as  the  other,  and  we  have 

P  :  P'  : :  a\/l   :  0^64, 
whence  P'  =  4  x  P. 

Thus  we  see  that  small  magnets  are  stronger  proportionally  than 
large  ones. 

The  most  powerful  permanent  magnet  yet  constructed,  accord- 
ing to  Gordon,  weighs  about  110  pounds,  and  sustains  ten  times 
its  own  weight.  It  is  said  that  small  magnets  have  been  con- 
structed which  would  carry  twenty-five  times  their  own  weight. 

602.  The   Declination  Compass. — This  instrument  con- 
sists of  a  magnetic  needle  suspended  in  the  centre  of  a  cylindrical 
brass  box  covered  with  glass ;  on  the  bottom  of  the  box  within  is 
fastened  a  circular  card,  divided  into  degrees  and  minutes,  from 
0°  to  90°  on  the  several  quadrants.     On  the  top  of  the  box  are  two 


382  MAGNETISM. 

uprights,  either  for  holding  sight-lines  or  for  supporting  a  small 
telescope,  by  which  directions  are  fixed.  The  quadrants  on  the 
card  in  the  box  are  graduated  from  that  diameter  which  is  verti- 
cally beneath  the  line  of  sight. 

When  the  axis  of  vision  is  directed  along  a  given  line,  the 
needle  shows  how  many  degrees  that  line  is  inclined  to  the  mag- 
netic meridian.  In  order  that  the  angle  between  the  line  and  the 
geographical  meridian  may  be  found,  the  declination  of  the  needle 
for  the  place  must  be  known. 

603.  The  Mariner's  Compass. — In  the  mariner's  compass 
(Fig.  342)  the  card  is  made  as  light  as  possible,  and  attached  to  the 

needle,  so  that  the  north  and 
south  points  marked  on  the  card 
always  coincide  with  the  magnetic 
meridian.  The  index,  by  which 
the  direction  of  the  ship  is  read, 
consists  of  a  pair  of  vertical  lines, 
diametrically  opposite  to  each 
other,  on  the  interior  of  the  box. 
These  lines,  one  of  which  is  seen 
at  0,  are  in  the  plane  of  the  ship's 
keel.  Hence,  the  degree  of  the  card  which  is  against  either  of 
the  lines  shows  at  once  both  the  angle  with  the  magnetic  meridian 
and  the  quadrant  in  which  that  angle  lies. 

In  order  that  the  top  of  the  box  may  always  be  in  a  horizontal 
position,  and  the  needle  as  free  as  possible  from  agitation  by  the 
rolling  of  the  ship,  the  box,  B,  is  suspended  in  gimbals.  The 
pivots,  A,  Ay  on  opposite  sides  of  the  box,  are  centred  in  the  brass 
ring,  (7,  J9,  while  this  ring  rests  on  an  axis,  which  has  its  bear- 
ings in  the  supports,  E,  E.  These  two  axes  are  at  right  angles  to 
each  other,  and  intersect  at  the  point  where  the  needle  rests  on  its 
pivot.  Therefore,  whatever  position  the  supports,  E,  E,  may  have, 
the  box,  having  its  principal  weight  in  the  lower  part,  maintains 
its  upright  position,  and  the  centre  of  the  needle  is  not  moved  by 
the  revolutions  on  the  two  axes. 

On  account  of  the  dip,  which  increases  with  the  distance  from 
the  equator,  and  is  reversed  by  going  from  one  hemisphere  to  the 
other,  the  needle  needs  to  be  loaded  by  a  small  adjustable  weight, 
if  it  is  to  be  used  in  extensive  voyages  to  the  north  or  south.  In 
north  latitudes  the  unmarked  end  must  be  heaviest ;  in  south 
latitudes,  the  marked  end. 

604.  The  Needle  Rendered  Astatic.— Though  magnetic 
intensity  increases  at  greater  distances  from  the  equator,  yet  the 


THEORY    OF    MAGNETISM.  383 

directive  power  of  the  compass  grows  more  feeble  in  approaching 
the  poles  of  dip,  because  the  horizontal  component  constantly 
diminishes,  and  at  the  poles  becomes  zero  (Art.  590).  A  needle  in 
such  a  situation,  in  which  the  earth's  magnetism  has  no  influence 
to  give  it  direction,  is  called  astatic.  The  compass  needle  is  astatic 
at  the  north  and  south  poles  of  dip.  And  the  dipping  needle  may 
be  rendered  astatic  at  any  place  by  setting  its  plane  of  rotation 
perpendicular  to  its  line  of  dip  at  that  place  ;  for  then  there  will 
remain  no  component  of  the  magnetic  force  in  the  only  plane  in 
which  the  needle  is  at  liberty  to  move  (Art.  578). 

The  needle  may  be  rendered  astatic  by  placing  a  magnet  above 
or  below  it,  with  its  axis  in  the  magnetic  meridian,  its  marked 
pole  pointing  north,  and  at  such  a  distance  from  the  needle  as  to 
cause  it  to  take  a  position  perpendicular  to  the  plane  of  the  mag- 
netic meridian.  Under  such  conditions  the  needle  will  be  ex- 
tremely sensitive. 

A  compound  needle,  consisting  of  two  simple  needles  fixed 
upon  a  wire,  as  in  Fig.  343,  with  their  unlike  poles  opposed,  may 
be  suspended  in  any  of  the  usual  modes. 
If  the  needles  are  exactly  equal  in  all  re-  FlG-  343- 

spects  the  system  will  be  perfectly  astatic. 
The  condition  of  perfect  equality  in  all 
the  conditions  is  never  realized. 

This  method  of  liberating  a  magnetic 
needle  from  the  earth's  influence  is  of 
great  use  in  electro-magnetism. 

Needles  may  be  balanced  upon  very 
sharp  pivots,  the  needle  cap  being  either 
of  hard  brass,  or  a  jewel  properly  drilled  ; 
this  mode  of  mounting  is  adopted  in 
declination  and  marine  compasses. 

For  delicate  experiments  a  better  suspension  is  a  filament  of 
silk,  or  for  light  needles  a  spider's  thread.  This  is  called  unifilar 
suspension ;  if  two  fibres,  parallel  and  very  near  together,  are 
used,  to  determine  the  amount  of  torsion,  we  have  bifilar  suspen- 
sion. 

605.  Theory  of  Magnetism. — The  nature  of  the  agency 
called  magnetism  is  unknown.  Much  of  the  language  employed 
by  writers  on  the  subject  implies  that  there  exist  in  iron,  steel, 
&c.,  two  imponderable  fluids,  called  the  austral  and  boreal  mag- 
netisms ;  that  these  fluids  attract  each  other,  and  are  ordinarily 
mingled  and  neutralized,  so  that  no  magnetic  phenomena  appear ; 
and  that  in  every  magnet  the  two  fluids  have  been  separated  by 
the  inductive  influence  of  the  earth  or  of  another  magnet,  one 


384  MAGNETISM. 

fluid  manifesting  itself  at  one  pole,  and  the  other  at  the  other 
pole.  As  science  advances,  however,  these  views  seem  more  and 
more  crude  and  unsatisfactory.  Magnetism  is  now  regarded  by 
many  as  one  of  those  modes  of  molecular  motion  which  are  so 
difficult  of  investigation.  If  it  is  a  mode  of  motion,  then  it  may 
manifest  itself  as  a  force,  as  we  know  it  does.  It  will  be  seen  in 
the  discussion  of  Electro-magnetism  that  there  is  a  most  intimate 
connection  between  magnetism  and  electricity,  so  much  so  that 
the  former  is  generally  considered  as  only  a  particular  form  in 
which  the  latter  is  developed. 

Magnetism  differs  from  the  other  molecular  agencies — electri- 
city, light  and  heat — in  producing  no  direct  effect  on  any  of  our 
senses.  We  witness'  its  direct  effects  only  in  the  motion  which  it 
gives  to  certain  kinds  of  matter,  such  as  iron  and  steel. 


PART    VIII. 

FEICTIONAL    OE    STATICAL    ELECTRICITY. 

i 


CHAPTER    I. 

ELEMENTARY     PHENOMENA. 

606.  Definition. — The  name  Electricity,  from  the  Greek  word 
for  amber,  is  given  to  a  peculiar  agency,  which  causes  mutual 
attractions  or  repulsions  between  light  bodies,  and  which,  under 
proper  conditions,  also  produces  heat,  light,  sound  and  chemical 
decomposition. 

Lightning  and  thunder  are  familiar  illustrations  of  the  intense 
action  of  this  agency. 

607.  Common  Indications  of  Electricity. — If  amber,  seal- 
ing-wax, or  any  other  resinous  substance,  be  rubbed  with  dry 
woolen  cloth,  fur,  or  silk,  and  then  brought  near  the  face,  the 
excited  electricity  disturbs  the  downy  hairs  upon  the  skin,  and 
thus  causes  a  sensation  like  that  produced  by  a  cobweb.    When 
the  tube  is  strongly  excited,  it  gives  off  a  spark  to  the  finger  held 
toward  it,  accompanied  by  a  sharp  snapping  noise.     A  sheet  of 
writing-paper,  first  dried  by  the  fire,  and  then  laid  on  a  table 
and  rubbed  with  India-rubber,  becomes  so  much  excited  as  to 
adhere  to  the  wall  of  the  room  or  any  other  surface  to  which  it  is 
applied.     As  the  paper  is  pulled  up  slowly  from  the  table  by  one 
edge,  a  number  of  small  sparks  may  be  seen  and  heard  on  the 
under  side  of  the  paper.     In  dry  weather,  the  brushing  of  a  gar- 
ment causes  the  floating  dust  to  fly  back  and  cling  to  it. 

Such  electricity  is  called  Frictional,  from  the  usual  mode  of  its 
development,  as  distinguished  from  Galvanic,  which  results  from 
chemical  action.  The  former  is  often  called  statical,  and  the 
latter  dynamical  electricity. 

Bodies  are  said  to  be  electrically  excited  when  they  show  signs 
of  electricity  in  consequence  of  some  mechanical  action  performed 
upon  them,  as  in  the  experiments  already  described. 
25 


386      FRICTIONAL    OR    STATICAL    ELECTRICITY. 

A  body  is  electrified  when  it  receives  electricity,  by  communica- 
tion, from  another  body  already  excited  or  electrified. 

608.  Pendulum  Electroscope. — Attraction  or  repulsion  is 
the  most  delicate  test  of  the  presence  of  electricity,  and  instru- 
ments prepared  for  showing  these  effects  are  called  electroscopes. 

t  The  word  electrometer,  though  sometimes  used  in  the 

same  sense,  is  more  properly  defined  to  be  an  instru- 
^\       ment  for  measuring  the  quantity  of  electricity. 

The  pendulum  electroscope  (Fig.  344)  consists  of  a 
glass  standard,  supported  by  a  base,  and  bent  into  a 
hook  at  the  top,  from  which  is  suspended  a  pith 
ball  by  a  fine  silk  thread. 

When  an  electrified  body  is  brought  near  the  pith 
ball,  its  movement  away  from  or  toward  the  body 
indicates  the  presence  of  an  electrical  charge ;  if 
there  is  no  motion  of  the  ball  the  body  is  not 
charged. 

Certain  modifications  are  convenient  for  some 
purposes.  In  one  a  metallic  wire  with  a  gilt  ball  on 
the  top  has  a  pith  ball  suspended  by  the  side  of  it ; 
in  another,  two  pith  balls  are  suspended  side  by  side,  as  in  Fig. 
353. 

609.  Nature  of  Electricity. — The  real  nature  of  electricity 
is  unknown.     Though  it  is  in  most  treatises  spoken  of  as  a  fluid, 
of  exceeding  rarity,  and  more  rapid  in  its  movements  than  light, 
yet  the  prevailing  belief  at  the  present  day  is,  that  it  is  a  peculiar 
mode  of  vibratory  motion,  either  in  the  luminiferons  ether  which 
is  imagined  to  fill  all  space,  or  else  in  the  ordinary  matter  con- 
stituting the  bodies  and  media  about  us,  or  in  both  of  these. 
Electricity  is  brought  to  view  by  friction,  by  heat,  and  by  other 
agencies  which  are  calculated  to  cause  movements  in  matter,  rather 
than  to  bring  new  kinds  of  matter  to  light.     It  is  undoubtedly 
one  of  the  forms  of  force,  into  which  other  forces  may  be  trans- 
formed.   But  until  a  more  definite  wave-theory  or  force-theory  can 
be  constructed  than  exists  at  present,  it  is  comparatively  easy  to 
give  to  the  learner  an  intelligible  description  of  electrical  phe- 
nomena by  using  the  language  of  the  two-fluid  theory  of  Du  Fay. 
In  trying  to  give  a  statement  of  observed  facts  without  the  use  of 
these  hypothetical  terms,  it  is  necessary  to  employ  in  their  stead 
tedious  circumlocutions,   which  only  confuse   the  mind  of  the 
learner. 

610.  Du  Fay's  Theory. — According  to  this  theory,  the  two 
fluids  are  imagined  to  inhere  in  all  kinds  of  matter,  combined  with 


TWO    ELECTRICAL    STATES.  337 

each  other  and  neutralized.  In  this  condition,  they  afford  no 
evidence  of  their  existence.  But  they  can  in  several  ways  be 
separated  from  each  other ;  and  when  thus  separated,  they  give 
rise  to  electrical  phenomena. 

611.  The  Two  Electrical  States.— After  friction  between 
two  bodies,  the  electrical  condition  of  each  is  unlike  that  of  the 
other.     These  two  electrical  states  are  usually  called  the  positive 
and  the  net/ative,  terms  which  were  employed  by  Franklin  in  his 
theory  of  one  electric  fluid,  to  indicate  that  the  excited  body  has 
either  more  or  less  electricity  than  belongs  to  it  in  its  common 
unexcUcul  condition.     Du  Fay  uses  the  words  vitreous  and  resin- 
ous to  distinguish  the  two  electrical  conditions,  vitreous  corre- 
sponding to  the  positive,  and  resinous  to  the  negative.     It  is  very 
common  to  use  Du  Fay's  theory,  and  to  apply  Franklin's  terms, 
positive  and  negative,  to  the  two  kinds  of  electricity. 

The  student  must  ever  bear  in  mind  that  these  terms  are 
merely  convenient,  and  their  use  must  not  lead  to  an  acceptance 
of  the  theory  of  a  fluid. 

612.  The  Two  States  Developed  Simultaneously.— If 

bodies  are  rubbed  together,  the  two  electricities  are  separated, 
and  one  body  is  electrified  positively,  the  other  negatively.  For 
example,  glass  rubbed  with  silk  is  itself  positive,  and  the  silk  is 
negative.  But  the  same  substance  does  not  always  show  the  same 
kind  of  electricity,  since  that  depends  frequently  on  the  substance 
against  which  it  is  rubbed.  Dry  woolen  cloth  rubbed  on  smooth 
glass  is  negative,  but  on  sulphur  it  is  positive.  The  following 
table  contains  a  few  substances,  arranged  with  reierence  to  this. 
Any  one  of  them,  rubbed  with  one  that  follows  it,  is  positively 
electrified  itself,  aud  the  other  negatively : 

1.  Fur  of  a  cat.  7.  Silk. 

2.  Smooth  glass.  8.  Gum  lac. 

3.  Flannel.  9.  Resin. 

4.  Feathers.  10.  Sulphur. 

5.  Wood.  11.  India-rubber. 

6.  Paper.  12.  Gutta-percha. 

According  to  the  above  table,  silk  rubbed  on  smooth  glass  is 
negatively  excited  ;  but  rubbed  on  sulphur,  it  is  excited  positively. 
It  is  sometimes  found,  however,  that  the  previous  electrical  con- 
dition of  one  of  the  bodies  will  invert  the  order  stated  in  the  table. 
For  example,  if  silk,  having  been  rubbed  on  smooth  glass,  and 
therefore  being  negative,  should  then  be  rubbed  on  resin,  it  would 
probably  retain  its  negative  state,  and  the  resin  become  positively 
electrified,  contrary  to  the  order  of  the  table. 


388      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

The  mechanical  condition  of  the  surface  sometimes  changes 
the  order  of  the  two  electricities.  Thus,  if  glass  is  ground,  so 
as  to  lose  its  polish,  it  is  likely  to  be  negative  when  rubbed  with 
silk  ;  but  the  excitation  of  rough  glass  is  very  feeble. 

613.  Mutual  Action. — Bodies  electrified  in  different  way* 
attract,  and  in  the  same  way  repel  each  other.     Thus,  if  an  insu- 
lated pith  ball,  or  a  lock  of  cotton,  be  electrified  by  touching  it 
with  an  excited  glass  tube,  it  will  immediately  recede  from  the 
tube,  and  from  all  other  bodies  which  are  charged  with  positive 
electricity,  while  it  will  be  attracted  by  excited  sealing-wax,  and 
by  all  other  bodies  which  are  negatively  electrified.    If  a  lock  of 
fine  long  hair  be  held  at  one  end,  and  brushed  with  a  dry  brush, 
the  separate  hairs  will  become  electrified,  and  will  repel  each 
other.    In  like  manner,  two  insulated  pith  balls,  or  any  other 
light  bodies,  will  repel  each  other  when  they  are  electrified  the 
same  way,  and  attract  each  other  when  they  are  electrified  in 
different  ways. 

Hence  it  is  easy  to  determine  whether  the  electricity  developed 
in  a  given  body  is  positive  or  negative ;  for,  having  charged  the 
electroscope  with  excited  glass,  then  all  those  bodies  which,  when 
excited,  attract  the  ball,  are  negatively  charged,  while  all  those 
which  repel  it  are  positively  charged. 

614.  Conduction. — Electricity  passes  through  some  bodies 
with   the  greatest  facility;   through   others  with  difficulty,    or 
scarcely  at  all ;  and  others  still  have  a  conducting  power  inter- 
mediate between  the  two.     As  the  conducting  quality  exists  in 
different  substances  in  all  conceivable  degrees,  it  is  impossible  to 
draw  a  dividing  line  between  them,  so  as  to  arrange  all  good  con- 
ductors on  one  side,  and  all  poor  conductors  on  the  other.    The 
following  brief  table  contains  some  of  the  more  important  of  the 
two  classes,  in  the  order  of  conducting  power  : 

Good  conductors.  Poor  conductors  (or  insulators). 

The  metals,  Baked  wood, 

Charcoal,  Air,  the  gases, 

Plumbago,  Paper, 

Water,  damp  snow,  Silk,  wool,  hair,  feathers, 

Living  vegetables,  Glass,  precious  stones, 

Living  animals,  Wax, 

Smoke,  steam,  Sulphur, 

Moist  earth,  stones,  Parafnne, 

Linen,  cotton.  Lac,  amber,  the  resins. 

Good  conductors  are  usually  termed  conductors,  while  poor 
conductors  are  called  insulators.  The  less  the  conductivity  of  any 
substance  the  greater  is  its  insulating  power. 


MODES    OF    INSULATING.  389 

When  air  is  rarefied,  its  insulating  power  is  diminished,  and 
the  further  the  rarefaction  proceeds,  the  more  freely  does  electri- 
city pass.  Hence,  we  might  expect  that  it  would  pass  with  per- 
fect freedom  through  a  complete  vacuum.  It  is  found,  however, 
that  in  an  absolute  vacuum  electricity  cannot  be  transmitted  at 
all. 

Although  under  ordinary  conditions  the  arrangement  of  the 
above  table  is  correct,  yet  conductivity  varies  with  changes  of 
physical  state.  Glass  becomes  a  much  better  conductor  at  red 
heat  than  it  is  when  cold  ;  resins  become  better  conductors  with 
rise  of  temperature ;  metals  become  poorer  conductors  with  in- 
crease of  temperature. 

615.  Modes  of  Insulating. — Solid  insulating  supports  are 
usually  made  of  glass ;  and,  in  order  to  improve  their  insulating 
power,  they  are  sometimes  covered  with  shell-lac  varnish.    Insu- 
lating threads  for  pith  balls,   or  cords  for  suspending  heavier 
bodies,  are  made  of  silk.     The  best  insulator  for  suspending  any 
very  small  weight  is  a  single  fiber  of  silk,  a  hair,  or  a  fine  thread 
of  gum-lac.     In  order  to  perform  electrical  experiments,  the  air 
must  be  dry,  or  no  care  whatever  relating  to  apparatus  can  insure 
success ;   and   therefore,   in   a  room   occupied  by  an   audience, 
especially  if  the  weather  is  damp,  it  is  necessary  to  dry  the  air 
artificially  by  fires,  and  to  warm  all  glass  insulators  to  drive  oif 
the  film  of  moisture  which  condenses  upon  them.      If  the  air 
were  a  good  conductor,  it  is  probable  that  no  facts  in  this  science 
would  ever  have  been  discovered. 

If  an  iron  or  brass  rod  be  held  in  the  hand  and  rubbed  with 
silk,  the  rod  shows  no  sign  of  electricity,  the  electricity  excited  in 
the  rod  being  conveyed  away  pIGJ  g^g 

by  the   conducting  quality  of 

the  metal  and  the  human  body ;  "^!  :^>jm#'wmmm//MA 

but  if  the  metal  rod  be  insulated 

by  a  glass  or  ebonite  handle,  as  in  Fig.  345,  it,  as  well  as  the  rub- 
ber, will  give  signs  of  electricity  when  properly  tested. 

616.  Other  Modes  of  Developing  Electricity.— Cleavage 
of  many  minerals  causes  unlike  electrical  states  in  the  separated 
laminae;  this  is  especially  noticeable  in  the  case  of  mica.     Loaf 
sugar,  broken  in  the  dark,  becomes  slightly  luminous. 

Pressure  of  a  crystal  of  Iceland  spar  will  produce  an  electric 
charge. 

Tourmaline  when  heated,  or  cooled,  becomes  electrically  charged 
at  two  opposite  ends  of  the  crystal,  owing  to  the  change  of  tem- 
perature, and  not  to  the  particular  temperature  at  any  instant. 
One  end  of  the  crystal  will  be  positive,  and  the  other  negative. 


390      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

This  development  of  electricity,  in  the  case  of  tourmaline,  seems 
to  be  confined  to  changes  of  temperature  between  10°  C.  and 
150°  C. 

Pyroelectricity  is  the  name  given  to  the  charge  thus  developed, 
and  crystals  so  affected  are  said  to  be  pyroelectric. 


CHAPTER    II. 

ELECTRICAL    MACHINES.— LAW    OF  FORCE.— MODE    OF    DISTRI- 
BUTION. 

617.  The  Plate  Machine.— In  order  that  glass  may  be 
conveniently  subjected  to  friction  for  the  development  of  electri- 

FIQ.  346. 


city,  it  is  made  in  the  form  of  a  circular  plate,  and  mounted  on 
an  axis,  which  is  supported  by  a  wooden  frame,  and  revolved  by  a 
crank,  while  rubbers  press  against  its  surface.  Fig.  346  represents 


CYLINDER    MACHINE.  391 

one  of  the  many  forms  which  have  been  adopted.  The  crank,  M9 
gives  rotary  motion  to  the  plate,  P,  which  is  pressed  by  the  rub- 
bers, F,  F ;  this  pressure  is  equalized  by  their  being  placed  at  top 
and  bottom,  and  on  both  sides  of  the  glass.  The  prime  conduc- 
tor, C  C,  is  made  of  hollow  brass,  and  supported  by  glass  pillars. 
The  extremities  terminate  in  two  bows,  which  pass  around  the 
edges  of  the  plate,  and  present  to  it  a  few  sharp  points,  to  facili- 
tate the  passage  of  electricity.  But  all  other  parts  are  carefully 
rounded  in  cylindrical  and  spherical  forms,  without  edges  or 
points,  as  they  tend  to  dissipate  the  electricity.  The  glass,  as  it 
revolves  from  the  rubbers  to  the  points  of  the  prime  conductor,  is 
protected  by  silk  covers,  to  prevent  the  electricity  from  escaping 
into  the  air.  The  rubbers  are  made  of  soft  leather,  attached  to  a 
piece  of  wood  or  metal,  and  from  time  to  time  are  rubbed  over 
with  an  amalgam  of  1  part  zinc,  1  part  tin,  and  two  parts  mercury, 
powdered  and  mixed  with  grease  ;  or  with  the  bi-sulphuret  of  tin, 
which  is  one  of  the  best  exciters  on  glass.  The  diameter  of  the 
plate  varies  from  1J-  to  3  feet ;  but  in  some  of  the  largest  it  is  6 
feet,  and  two  plates  are  sometimes  mounted  on  one  axis. 

To  give  free  passage  of  the  negative  electricity  from  the  rub- 
bers to  the  earth,  a  chain,  D,  may  be  attached  to  the  wooden 
support,  while  its  other  end  lies  on  the  floor. 

It  is  necessary  to  dry  and  warm  all  insulators  of  the  machine 
to  prevent  escape  of  electricity  by  the  supports  ;  also  to  dry  the 
surrounding  air  (Art.  615).  As  the  electrical  charge  increases 
with  the  rapidity  of  the  rotation,  soon  the  accumulation  will  be 
so  great  as  to  cause  some  of  the  electricity  developed  upon  the 
plate  to  overcome  the  low  conductivity  of  the  glass  and  recombine 
with  the  opposite  electricity  of  the  rubber,  and  a  greater  charge 
than  this  can  not  be  maintained  by  any  increase  of  velocity  of 
rotation. 

618.  The  Cylinder  Machine.— In  many  electrical  machines 
of  the  smaller  sizes,  a  hollow  cylinder  is  employed,  having  a  length 
considerably  exceeding  its  diameter.     In  the  cylinder  machine, 
the  rubber  is  applied  to  one  side,  and  the  prime  conductor  receives 
the  fluid  from  the  opposite.     The  rubber  is  usually  mounted  on  a 
glass  pillar,  so  that  it  can  be  insulated,  whenever  it  is  desired. 

619.  The  Hydro-Electric  Machine. — It  was  discovered  in 
1840  that  a  steam-boiler  electrically  insulated  gave  out  sparks, 
and  that  the  steam  issuing  from  it  was  also  electrified.     Hence 
resulted  the  construction  of  the  hydro-electric  machine.     It  con- 
sists of  a  boiler  mounted  on  glass  pillars,  and  furnished  with  a  row 
of  jet-pipes  and  a  metallic  plate,  against  which  the  steam  strikes. 
The  prime  conductor,  to  which  the  steam-plate  is  attached,  is 


392      7RICTIONAL  OR  STATICAL  ELECTRIC    TY. 

electrified  positively,  and  the  boiler  itself  negatively.  Faraday 
ascertained  that  the  electricity  in  this  case  is  developed,  not  by 
evaporation  or  condensation,  but  by  the  friction  of  watery  particles 
in  the  jefcpipes,  perfectly  dry  steam  producing  no  charge.  That 
the  machine  may  act  with  energy,  it  was  found  necessary  to  make 
the  interior  of  the  jet-pipes  angular,  and  quite  irregular. 

A  machine  of  this  kind,  whose  boiler  was  6J  feet  long  and  3J 
feet  in  diameter,  gave  sparks  22  inches  in  length.  The  great 
amount  of  moisture  discharged,  and  the  necessary  and  trouble- 
some precautions  in  preparing  the  machine  for  use,  render  it 
valueless  for  any  practical  purpose. 

620.  The  Quadrant  Electrometer.— In  order  to  measure 
the  intensity  of  electricity  in  the  prime  con- 
ductor, there  is  set  upon  it,  whenever  desired, 
a  quadrant  electrometer  (Fig.  347).  This  con- 
sists of  a  pillar,  d,  about  six  inches  high,  having 
a  graduated  semicircle,  c,  attached  to  one  side, 
and  a  delicate  rod  and  ball,  a,  suspended  from 
the  centre  of  the  semicircle.  As  the  conductor 
becomes  electrified,  the  rod  is  repelled  from  the 
pillar,  and  the  arc  passed  over  indicates  rudely 
the  degree  of  electrical  intensity. 

621.  First  Phenomena  of  the  Ma- 
chine.— When  an  electrical  machine  is  skill- 
fully fitted  up,  and  works  well,  there  is  first 
perceived,  on  turning  it,  a  crackling  sound;  and  then,  on  bring- 
ing the  knuckles  toward  the  prime  conductor,  a  brilliant  spark 
leaps  across,  causing  a  sharp  pricking  sensation.  If  the  room  be 
darkened,  brushes  of  pale  light  are  seen  to  dart  off  continually 
from  the  most  slender  parts  of  the  prime  conductor,  with  a  hiss- 
ing or  fluttering  noise,  while  circles  of  light  snap  along  the  glass 
between  the  rubbers  and  the  edges  of  the  covers.  When  electricity 
is  escaping  plentifully  from  the  machine,  a  person  standing  near 
also  perceives  a  peculiar  odor,  which  is  that  of  ozone,  and  which 
seems  always  to  accompany  the  development  of  electricity. 

Therefore,  at  least  four  of  the  senses  are  directly  affected  by 
this  remarkable  agency,  while  magnetism  affects  none  of  them. 

The  phenomena  of  repulsion  of  like  and  attraction  of  unlike 
electricities,  are  well  shown  by  the  machine.  A  skein  of  thread 
or  a  tuft  of  hair,  suspended  from  the  prime  conductor,  will,  as 
soon  as  the  plate  is  revolved,  spread  into  as  wide  a  space  as  possi- 
ble, by  the  repellency  of  the  fibers  which  are  electrified  alike. 
Melted  sealing-wax  is  thrown  off  in  fine  threads,  and  dropping 
water  is  diverged  into  delicate  filaments.  Even  air,  on  those  parts 


COULOMB'S  TORSION  BALANCE. 


393 


FIG.  348. 


of  the  prime  conductor  which  are  most  strongly  charged,  becomes  so 
self-repellent  as  to  fly  off  in  a  stream  of  wind,  which  is  plainly  felt. 

On  the  other  hand,  light  bodies,  when  brought  toward  the 
machine  while  in  action,  instantly  fly  to  the  prime  conductor ;  for 
that  is  positive,  but  the  nearer  sides  of  the  other  bodies  are  made 
negative  by  induction. 

The  difference  between  substances  as  to  their  conducting 
quality  is  readily  perceived  by  setting  the  quadrant  electrometer 
on  the  prime  conductor,  raising  the  index  by  turning  the  plate, 
and  then  touching  the  prime  conductor  with  the  remote  end  of 
the  body  to  be  tried.  If  an  iron  rod,  or  even  a  fine  iron  wire,  be 
thus  applied,  the  index  will  fall  instantly  ;  a  long  dry  wooden  rod 
will  cause  it  to  descend  slowly,  while  a  glass  rod  will  produce  no 
effect  at  all.  These  experiments  show  that  iron  is  a  good  con- 
ductor, wood  an  imperfect  conductor,  and  glass  a  very  poor  con- 
ductor. 

622.  Coulomb's  Torsion   Balance. — When  a  long  fine 
wire  is  stretched  by  a  small  weight,  its  elasticity  of  torsion  is  a 
very  delicate  force,  which  is  successfully  employed  for  the  measure- 
ment of  other  small  forces.     When  such  a  wire  is  twisted  through 
different  angles,  the  force  of  torsion  is 

found  to  vary  as  the  angle  of  torsion  ; 

it  is  therefore  easy  to  measure  the  force 

which   is  in  equilibrium  with  torsion. 

The  torsion  balance  is  represented  in 

Fig.   348.     The  needle  of  lac,  n  o,  is 

suspended  by  a  very  fine  wire  from  a 

stem  at  the  top  of  the  tube  d.     The  cap 

of  the  tube,  e,  is  a  graduated  circle, 

whose  exact  position  is  marked  by  the 

index,  a.     The  stem   from  which  the 

wire  hangs  is  held  in  place  in  the  centre 

of  the  cap  by  friction,  but  can  be  turned 

round  so  as  to  place  the  needle  in  any 

direction  desired.     At  the  end  of  the  lac 

needle  is  a  small  disk  of  brass-leaf,  n, 

and  by  its  side  a  gilt  disk,  m,  connected 

with  the  handle,  r,  by  the  glass  rod,  i. 

This  apparatus  is  suspended  in  the  glass  cylinder,  covered  with  a 

glass  plate,  on  the  centre  of  which  the  tube  d  is  fastened.     There 

is  a  graduated  circle   around  the  cylinder  on  the  level  of  the 

needle. 

623.  Law  of  Electrical  Force  as  to  Distance.— Adjust- 
ment is  now  made  by  turning  the  stem  so  that,  while  the  wire  is 


394      PRICTIONAL  OR  STATICAL  ELECTRICITY. 

in  its  natural  condition,  the  disk,  n,  touches  the  disk,  m,  and  is  at 
zero,  and  the  index  at  top  also  at  zero  on  the  circle  e.  Let  a 
minute  charge  of  electricity  be  communicated  to  m,  and  it  will 
repel  n,  and  cause  it,  after  a  few  oscillations,  to  settle  at  a  certain 
distance  —  suppose  for  instance,  at  36°.  The  circles  is  now  turned 
in  the  opposite  direction,  until  the  needle  is  brought  within  18° 
of  the  ball  m.  In  order  to  bring  it  thus  near,  the  index  has  to  be 
turned  126°,  which  added  to  the  18°,  makes  the  whole  torsion 
144°,  or  four  times  as  great  as  before.  Therefore,  at  one-half  the 
distance  there  is  four  times  the  repulsion.  In  like  manner,  it  is 
found  that  at  one-third  the  distance  there  is  nine  times  the  repul- 
sion. Hence  the  law  that  for  two  given  charges, 

Electrical  repulsion  varies  inversely  as  the  square  of  the  dis- 
tance. 

Ill  a  manner  somewhat  similar  to  the  foregoing,  it  was  conclu- 
sively proved  by  Coulomb  that  electrical  attraction  obeys  the  same 
law  of  distance,  though  there  is  more  practical  difficulty  in  per- 
forming the  experiments.  But  if  the  electrified  body  m  is  placed 
outside  of  the  circle  described  by  n,  so  that  the  latter  is  allowed 
to  vibrate  both  to  the  right  and  left,  the  square  of  the  number  of 
vibrations  in  a  given  time  becomes  a  measure  of  the  attractive 
force,  as  in  the  case  of  the  pendulum  (Art.  163). 

If  the  charge  upon  m  be  doubled,  that  upon  n  remaining  un- 
changed, the  angle  of  torsion  will  be  double  that  found  above  in 
each  case  ;  and  if  we  now  treble  the  charge  upon  n,  the  angle  of 
torsion  will  be  six  times  as  great  as  at  first.  Hence  we  learn  that 
the  force  of  attraction  or  repulsion,  for  any  given  distance,  is  pro- 
portional to  the  product  of  the  two  charges.  Calling  the  charges 
q  and  q'  respectively,  the  distance  between  them  d,  and  force  ex- 
erted F,  we  have 


an  expression  similar  to  that  found  in  Art.  574. 

624.  Unit  of  Electricity.  —  The  expression  for  the  force  of 
repulsion  between  two  quantities,  q  and  q',  of  like  electricity,  at 

the  distance  I  from  each  other,  was  found  above  to  be/*  =  jj-  ;  if 

q2 
we  make  q  =  q'  we  have/  =  i«    If  ^  be  taken  equal  to  one  unit 

of  length  in  any  system  which  we  may  choose  to  adopt,  f  equal 
to  one  unit  of  force,  and  the  separating  medium  be  air,  then  q  will 
be  the  unit  of  electricity.  In  the  C.  G.  S.  system  I  =  one  cen- 
timetre, and  /  =  one  dyne  ;  therefore  the  unit  of  electricity  in 
tlii  system  is  that  quantity  of  electricity  which  ivill  repel  an 


STATICAL    ELECTRICAL    CHARGE.  395 

equal  quantity,  at  the  distance  one  centimetre)  with  a  force  of  one 
dyne. 

Suppose  two  spheres,  very  small  as  compared  with  their  dis- 
tance from  each  other,  to  be  charged  positively,  one  having  a 
charge  three  times  as  great  as  that  of  the  other,  and  that  when 
placed  five  centimetres  apart  they  repel  with  a  force  of  three 
dynes,  then  we  have  /  x  I2  =  q  X  3  q,  whence  3  x  52  =  3  q*  and 
q  =  5 ;  hence  one  spliere  was  charged  with  5  units  of  electricity, 
and  the  other  with  15  units. 

625.  Waste  of  Electricity  from  an  Insulated  Body. — 

In  making  accurate  investigations  like  the  foregoing,  in  which 
considerable  time  is  necessarily  occupied,  a  difficulty  arises  from 
the  loss  of  the  electrical  charge.  The  first  and  most  obvious 
source  of  waste  is  the  moisture  in  the  air,  which  conducts  away 
the  fluid  ;  but  this  may  be  nearly  avoided  by  setting  into  the 
cylinder  a  cup  of  dry  lime,  or  other  powerful  absorbent  of  moisture, 
as  represented  in  the  figure.  A  second  is  the  imperfect  insulation 
afforded  by  even  the  most  perfect  non-conductors.  A  third  is  the 
mobility  of  the  air,  whose  particles,  when  they  have  touched  the 
electrified  body,  and  become  charged,  are  repelled,  taking  away 
with  them  the  charge  they  have  received.  The  loss  in  these  ways 
is  very  slight,  when  the  charge  is  small,  and  allowance  can  be 
made  for  it  with  a  good  degree  of  accuracy.  But  when  bodies  are 
highly  charged,  they  lose  their  electricity  at  a  rapid  rate. 

626.  A  Statical  Electrical  Charge  Lies  at  the  Sur- 
face.— This  is  proved  in  many  ways.     A  hollow  ball,  no  matter 
how  thin,  will  receive  as  large  a  quantity  of  electricity  as  a  solid 
one.     Hence  it  is  that  the  prime   conductor  of    the  electrical 
machine,  and  metallic  articles  of  electrical  apparatus  generally, 
are  made  of  sheet  brass,  for  the  sake  of  lightness. 

Let  a  dish  a  (Fig.  349),  be  made  of  two  brass  rings  and  cam- 
bric sides  and  bottom,  with  an  insulating 
handle,  #,  attached  to  the  larger  ring.  If 
this  vessel  be  charged  with  electricity,  the 
charge  is  found  on  the  outside ;  turn  it 
over  quickly,  so  as  to  throw  it  the  other 
side  out,  and  the  charge  is  instantly  found 
on  the  outside  again,  and  none  on  the  inside.  It  may  be  inverted 
several  times  with  the  same  result,  before  the  charge  becomes  too 
feeble  to  be  perceived. 

If  an  insulated  hollow  metallic  sphere,  or  a  cylinder  of  wire 
gauze,  be  charged  and  its  interior  surface  be  tested,  no  charge  will 
be  detected.  To  make  this  test,  a,  proof  plane,  consisting  of  a  gilt 


396      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

disk  insulated  by  a  slender  rod  of  lac,  is  touched  to  the  interior 
surface,  and  is  then  applied  to  a  delicate  torsion  balance ;  as  no 
motion  of  the  needle  ensues,  it  is  proof  that  no  charge  was 
present. 

Another  proof  that  the  charge  occupies  only  the  outside  sur- 
face is  that  the  intensity  diminishes  as  the  surface  is  enlarged, 
while  the  mass  of  the  conductor  remains  the  same.  A  metallic 
ribbon  rolled  upon  an  insulated  cylinder  may  be  unrolled,  and 
thus  the  surface  enlarged  to  any  extent.  An  electroscope  stand- 
ing on  the  instrument  will  fall  as  the  ribbon  is  unrolled,  and  rise 
when  it  is  again  rolled  up. 

It  will  be  shown  in  Art.  636,  that  a  charge  may  be  induced 
upon  the  interior  surface  of  a  hollow  conductor,  but  it  will  be 
observed  that  this  fact  is  no  contradiction  of  those  given  above. 

A  charge  moving  from  one  point  to  another  passes  through  the 
substance  of  the  conductor  and  not  over  its  surface  merely.  If  a 
powerful  charge  be  sent  through  a  narrow  strip  of  gold  leaf  it  will 
melt,  and  perhaps  vaporize,  the  metal ;  a  similar  charge  passed 
through  a  gold  wire  whose  surface  equals  that  of  the  gold  leaf  will 
produce  no  such  effect,  owing  to  the  vastly  greater  cross-section  of 
the  wire. 

627.  Potential. — The  difference  of  electrical  condition  of 
two  bodies  which  causes  a  transfer  of  electricity  from  one  to  the 
other,  either  through  a  good  conductor  or  across  a  poorly-con- 
ducting medium,  is  called  difference  of  potential.     The  body,  A , 
from  which  electricity  is  transferred,   is  said  to   have  a  high 
potential  relatively  to  the  lower  potential  of  the  body,  B,  to  which 
the  transfer  is  made. 

The  measure  of  difference  of  potential  is  the  amount  of  work 
which  would  be  done  in  moving  a  unit  of  electricity  from  B  to  A, 
in  a  direction  contrary  to  that  in  which  a  free  unit  would  move. 
The  potential  of  the  earth  is  taken  as  the  standard  from  which  to 
measure  these  differences,  and  is  called  zero  ;  hence  the  difference 
of  potential  between  a  body,  or  point,  and  the  earth,  is  called  the 
potential  of  the  body,  or  point. 

In  Art.  36,  we  have  work  =  force  x  distance  ;  hence,  calling 
Va  and  Vb  the  potentials  at  A  and  B  respectively,  A  B  their 
distance  apart,  and /the  average  force  of  repulsion,  we  have 

f  x,AB  =  Vm-  Vb,  whence /=Z^^». 

When  A  and  B  are  near  together,  the  force  of  repulsion  at  all 
points  of  the  line  joining  them  will  be  practically  constant. 

628.  Equipotential  Surfaces.— Suppose  A  (Fig.  350)  to 


CAPACITY.  .    397 

be  an  electrical  charge,  concentrated  at  a  point,  and  let  B  represent} 

a  unit  of  electricity  at  a  distance  r.     The  difference  of  potential 

will  be  measured  by  the  work  required 

to  move  B,  through  the  distance  r,  to-  Fia.  350. 

wards  A ;  this  work  will  be  the  same 

wherever  B  may  be  placed   upon  the 

sphere  having  r  for  its  radius ;  hence 

the  definition, 

An  equipotential  surface,  with  respect 
to  an  electrified  point  A,  is  a  surface 
such  that  the  work  required  to  transfer 
a  unit  charge  from  any  point  of  it  to  the 
point  A  will  be  constant. 

At  any  point  of  the  sphere  O,  the  potential  will  be  lower  than 
upon  any  point  of  the  sphere  B,  since  more  work  will  be  done  in 
carrying  the  unit  charge  from  C  to  A,  than  from  B  to  A.  If  C 
be  so  far  from  B  as  to  cause  an  expenditure  of  one  unit  of  work 
in  moving  unit  charge  from  C  to  B,  there  will  be  a  unit  differ- 
ence of  potential  between  these  two  equipotential  surfaces. 

The  distances  A  B,  B  C,  CD,  &c.,  form  an  increasing  series, 
since  the  force  of  repulsion  decreases  while  the  radii  increase,  and 
hence  a  unit  charge  must  be  moved  through  increasing  distances 
to  produce  the  unit  of  work. 

An  equipotential  surface  about  two  electrified  points,  near  each 
other,  would  not  be  spherical  ;  and  if  a  number  of  electrified; 
bodies,  or  points,  constitute  an  electrical  centre,  the  equipotential 
surface  with  respect  to  them  would  be  irregular. 

The  unit  of  work,  usually  taken  in  this  connection,  is  the  work 
done  in  overcoming  a  resistance  of  one  dyne  through  a  distance 
of  one  centimetre,  and  is  called  an  erg. 

629.  Capacity. — The  number  of  units  of  electricity  which 
must  be  imparted  to  a  body  to  raise  its  potential  from  zero  to 
unity  measures  the  relative  electrical  capacity  of  the  body. 

A  body  that  requires  only  a  unit  of  electricity  to  raise  its 
potential  from  0  to  1,  is  said  to  have  unit  capacity.  A  unit  of 
electricity  communicated  to  a  sphere  of  one  centimetre  radius, 
will  raise  its  potential  from  0  to  1 ;  such  a  sphere  has  therefore 
unit  capacity. 

The  capacities  of  spheres  are  proportional  to  their  radii. 

In  the  above  definitions  no  external  disturbing  electrical  forces 
are  considered.  The  capacity  of  a  conductor  is  increased  by 
bringing  near  it  a  charge  of  opposite  kind ;  for  the  potential  of 
the  charged  conductor,  in  such  case,  is  the  difference  of  potential 
due  to  its  own  charge  and  that  due  to  the  charge  of  opposite  kind  ; 


398      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

hence  a  greater  charge  must  be  given  the  conductor  to  raise  its 
potential  from  0  to  1. 

630.  Distribution  of  a  Charge  on  the  Surface. — Statical 
electricity  resides  at  the  surface  of  a  body,  as  we  have  seen,  but  is 
not  uniformly  diffused  over  it,  except  in  the  case  of  the  sphere. 
In  general,  the  more  prominent  the  part,  and  the  more  rapid  its 
curvature,  the  more  intensely  is  the  fluid  accumulated  there. 

In  a  long  slender  rod  the  density  is  greatest  at  the  ends,  nearly 
the  whole  charge  being  collected  at  these  points.  On  a  sphere, 
not  influenced  by  induction,  the  density  is  uniform,  as  illustrated 
in  Fig.  351,  the  dotted  line  denoting  by  its  constant  distance  from 
the  surface  the  uniform  distribution  of  the  charge. 

Fig.  352  represents  the  varying  density  upon  an  ellipsoid. 

FIG.  851.  FIG.  352. 


The  two  ellipsoids  are  similar,  and  the  ellipsoidal  shell  included 
between  them  represents  the  densities  at  various  points.  In  this 
case  the  densities  at  any  two  points  of  the  ellipsoid  are  nearly 
proportional  to  the  diameters  through  those  points. 

The  student  must  remember  that  the  charge  does  not  form  a 
layer  upon  the  body  in  any  sense  whatever,  and  that  the  above 
figures  are  given  merely  to  aid  the  memory  in  retaining  the  law 
of  distribution. 

631.  Surface  Density. — The  number  of  units  of  electricity 
per  unit  area  of  surface  is  called  the  surface  density  at  the  point. 
Let  Q  be  the  quantity  of  electricity  on  any  given  area  8,  then  the 
surface  density,  called  p,  will  be  expressed  by  the  equation 

,-t- 

In  dry  air  the  limit  of  charge  is  given  as  about  20  units  per 
square  centimetre.  If  the  distribution  of  charge  be  uniform,  as 
in  the  case  of  a  sphere  removed  from  the  influence  of  electrical 
inductions,  the  density  is  found  by  dividing  the  total  charge  Q  by 

the  area  of  surface,  4  n  r2,  giving  p  =.          2;  hence,  if  equal 

charges  be  given  to  two  spheres  of  different  radii,  the  densities 
will  be  inversely  as  the  squares  of  the  radii. 


ATMOSPHERIC    PRESSURE.  399 

Two  spheres,  at  some  distance  apart  and  connected  by  a  tine 
wire,  will  divide  a  common  charge  imparted  to  them  in  the 
proportion  of  their  capacities,  that  is  to  say,  in  the  propor- 
tion of  their  radii.  Suppose  the  radius  of  one  to  be  r,  and  of 
the  other  2  r,  then  their  charges  will  be  as  r  to  2  r.  The  densities 
will  be  found  by  dividing  the  charges  by  the  respective  surfaces, 

r      2  r 

which  are  as  r8  to  (2r)2 ;  hence  we  have  -2 :  ^ — ^  : :  2  r  :  r,  or  the 

T       \6    T) 

densities  are  inversely  as  the  radii.  If  the  diameter  of  the  smaller 
sphere  is  made  less  and  less,  its  capacity  grows  less  also,  and  the 
potential  grows  greater  and  greater,  till  at  the  limit,  when  the 
sphere  becomes  a  point,  the  potential  is  so  great  as  to  cause  a  total 
discharge,  and  prevent  any  accumulation  of  charge  upon  it. 

632,  The  Charge  Held  on  the  Surface  by  Atmospheric 
Pressure. — The  mutual   repellency,  which  forces  the  electric 
charge  to  remain  upon  the  surface  of  the  conductor,  tends  to 
make  it  escape  in  all  directions  from  that  surface ;  and  it  is  the 
air  alone  which  prevents.     For  if  one  extremity  of  a  charged  and 
insulated  conductor  extends  into  the  receiver  of  an  air-pump,  the 
charge  is  dissipated  by  degrees,  as  the  receiver  is  exhausted  ;  and 
when  the  exhaustion  is  as  complete  as  possible,  the  most  abundant 
supply  from  the  machine  fails  to  charge  the  conductor. 

If  the  vacuum  is  made  as  perfect  as  possible  no  dissipation  of 
charge  occurs,  since  electricity  will  not  pass  through  a  vacuum 
(Art.  614). 

This  limit  cannot  be  reached  by  the  use  of  even  the  best  air- 
pump  alone,  but  is  attainable  in  other  ways. 

As  the  atmospheric  pressure  is  limited  to  about  15  Ibs.  per 
square  incn,  so  the  amount  of  charge  is  limited  which  can  be  re- 
tained on  a  conductor  of  given  form.  Hence  the  reason  for  the 
well-known  fact  that  the  prime  conductor  receives  all  the  charge 
which  it  is  capable  of  retaining  in  one  or  two  turns  of  the  machine. 
All  that  is  gained  orer  and  above  this,  by  continuing  to  turn,  flies 
off  through  the  air. 

633.  Rotation  by  Unbalanced  Pressure.— As  the  electric 
charge  on  the  surface  of  a  body  presses  outward  in  all  directions, 
wherever  it  escapes  from  a  point,  there  the  pressure  is  removed  ; 
consequently,  on  the  opposite  part  there  is  unbalanced  pressure. 
Therefore,  if  the  body  is  delicately  suspended,  and  one  or  more 
points  are  directed  tangentially,  the  unbalanced  pressure  will  cause 
rotation  in  the  opposite  direction,  just  as  Barker's  mill  rotates  by 
the  unbalanced  pressure  of  water.     Electrical  wheels  and  orreries 
are  revolved  in  this  way. 

A  windmill  may  also  be  revolved  by  the  stream  of  air  issuing 
from  a  stationary  point  attached  to  the  prime  conductor  (Art.  621). 


400      FRICTIONAL  OR  STATICAL  ELECTRICITY. 


CHAPTER    III. 

ELECTRICITY    BY    INDUCTION.— LEYDEN  JAR. 

634.  Elementary  Experiment. — When  an  electrified  body 
is  placed  near  one  which  is  unelectrified,  but  not  near  enough  to 
cause  discharge  between  them,  the  natural  electricities  of  the  latter 
are  decomposed,  one  being  attracted  toward  the  former,  the  other 
repelled  from  it.  Thus  the  ends  become  electrified  by  the  influence 
of  the  first  body,  without  receiving  any  electricity  from  it.  Let 
A  (Fig.  353)  be  charged  with  positive  electricity,  and  let  the 

FIG.  353. 


A  A  I  A  A 


insulated  conductor,  B  C,  be  furnished  with  several  electroscopes, 
as  represented.  Those  at  the  end  B  will  diverge  more  widely 
than  those  which  follow,  in  the  direction  B  C,  until  a  point  is 
reached,  somewhat  less  than  half  the  distance  from  B  to  C,  where 
there  is  no  sign  of  electricity ;  passing  this  neutral  point  the 
electroscopes  diverge  more  and  more  till  the  maximum  effect  is 
again  reached  at  C.  By  taking  off  small  quantities  with  the  proof 
plane  it  is  found  that  B  is  charged  negatively  and  6' positively, 
and  also  that  the  negative  charge  at  B  is  of  greater  density  than 
the  positive  at  C,  owing  to  the  attraction  of  unlike  electricities  at 
A  and  B  through  the  short  distance  A  B,  being  greater  than  the 
repulsion  between  the  like  charges,  at  A  and  C,  through  the 
greater  distance  A  C. 

Eemove  the  bodies  to  a  distance  from  each  other,  and  B  0 
returns  to  its  unelectrified  condition ;  bring  them  near  again,  and 
it  is  electrified  as  before.  As  this  electrical  state  is  induced  upon 


ACTIONS    AND    REACTIONS. 


401 


the  conductor  by  the  electrified  body  in  its  vicinity,  without  any 
communication  of  electricity,  it  is  said  to  be  electrified  by  induc- 
tion. If  A  is  first  charged  with  the  negative  electricity,  the  two 
electricities  of  B  (7 will  be  arranged  in  reversed  order;  the  posi- 
tive will  be  attracted  to  the  nearest  end,  the  negative  repelled  to 
the  farthest. 

Electrical  induction  is  exactly  analogous  to  magnetic  induc- 
tion ;  the  opposite  kind  is  developed  at  the  nearer  end,  and  the 
like  kind  at  the  remote  end. 

635.  Successive  Actions  and  Reactions. — If  A  is  itself 
an  insulated  conductor,  the  foregoing  is  not  the  entire  effect :  for 
a  reflex  influence  is  exerted  by  the  electricity  in  the  nearer  end  of 
the  conductor.     Let  A  have  a  positive  charge,  as  at  first.     After 
the  negative  electricity  is  attracted  to  the  nearer  end  of  B  C,  it  in 
turn  attracts  the  positive  charge  of  A,  and  accumulates  it  on  the 
nearest  side,  leaving  the  remote  side  less  strongly  charged  than 
before.     This  is  shown  by  electroscopes  attached  to  the  opposite 
sides  of  A.     The  charge  of  A,  being  now  nearer,  will  exert  more 
power  on  B  C,  separating  more  of  its  original  electricities,  and 
thus  making  the  nearest  end  more  strongly  negative  and  the 
remote  end  more  strongly  positive  than  before;  and  this  new 
arrangement  of  fluids  in  B  C  causes  a  second  reaction  upon  A,  of 
the  same  kind  as  the  first.     Thus  an  indefinite  diminishing  series 
of  adjustments  takes  place  in  a  single  moment  of  time. 

636.  Induced   Charge  within  a  Hollow  Conductor. — 

Let  A  (Fig.  354),  represent  a  section  through  an  insulated,  hollow, 
metallic  cylinder  upon  an  insulating  stand.  In- 
troduce within  it  the  charged  ball  B,  suspended 
by  a  long  silk  thread  for  insulation.  Connect 
the  outer  surface  of  the  cylinder,  by  a  wire  con- 
ductor, with  a  delicate  electroscope.  If  B  is 
charged  positively  it  will  induce  upon  the  inner 
surface  of  A  an  equal  quantity  of  negative  elec- 
tricity, while  the  outer  surface  will  become 
positively  charged.  When  B  is  removed  the 
cylinder  will  return  to  its  neutral  state.  As  B 
is  lowered  into  A  the  electroscope  will  indicate 
a  gradually  increasing  charge  upon  the  outer 
surface,  until  the  ball  reaches  a  point  two  or 
three  inches  below  the  mouth  of  the  cylinder, 
after  passing  which  a  further  descent  of  the  ball 
produces  no  change.  If  the  ball  be  allowed  to 
touch  the  bottom  of  A,  no  change  in  the  indications  of  the  electro- 
scope will  appear,  although  the  whole  charge  of  B  will  be  com- 
26 


FIG.  354. 


402      FRICTIONAL    OR    STATICAL    ELECTRICITY. 

municated  to  that  induced  upon  the  inner  surface  of  A,  exactly 
neutralizing  it,  and  on  withdrawing  B  it  will  be  found  wholly 
discharged ;  this  constancy  of  the  indication  of  the  electroscope 
shows  that  the  induced  charge  upon  the  outside  of  A  is  exactly 
equal  to  the  original  charge  upon  B. 

To  vary  the  experiment,  let  A  be  charged  positively,  the  whole 
of  which  charge  will  reside  upon  the  outer  surface. 

If  now  the  ball  B,  connected  to  earth  by  a  suspending  wire 
conductor,  be  lowered  into  it,  B  will  be  charged  negatively  by  the 
inductive  action  of  A,  a  portion  of  whose  positive  charge  is  thus 
transferred  to  the  inner  surface  by  the  reciprocal  action- of  the 
negative  induced  upon  B. 

637.  Division  of  the  Conductor.— Suppose  that  before  the 
experiment  (Art.  034)  begins,  B  C  is  in  two  parts  with  ends  in 
contact ;  the  entire  series  of  mutual  actions  takes  place  as  already 
described.    Now,  while  A  remains  in  the  vicinity,  let  the  parts  of 
B  C  be  separated  ;  then  the  negative  electricity  is  secured  in  the 
nearest  half,  and  the  positive  in  the  other.     And  if  A  is  now 
removed,  each  charge  diffuses  over  that  half  of  the  original  con- 
ductor upon  which  it  was  induced. 

Thus  each  kind  of  electricity  can  be  completely  separated  from 
the  other  by  means  of  induction. 

Here  we  find  a  marked  difference  between  magnetism  and 
frictional  electricity.  The  electricities  may  be  secured  in  their 
separate  state,  one  in  one  conductor,  the  other  in  another.  In 
magnetism  this  is  not  possible  ;  for  when  an  iron  bar  is  magnet- 
ized, and  then  broken,  each  kind  of  magnetism  is  found  in  each 
half  of  the  bar  (Art.  572)  ;  at  the  point  of  division  both  polari- 
ties exist,  and  as  soon  as  the  bar  is  broken,  they  manifest  them- 
selves there  as  strongly  as  at  the  extremities. 

638.  Effect  of  Lengthening  the  Conductor.— If  the  con- 
ductor, B  C,  is  lengthened,  the  accumulation  on  the  adjacent  parts 
of  the  two  bodies  is  somewhat  increased.    The  positive  electricity 
which,  at  the  remote  end  of  the  shorter  conductor,  operated  in 
some  degree  by  its  repulsion  of  the  charge  upon  A  and  its  attrac- 
tion for  that  upon  B  to  prevent  accumulation  on  the  nearest  side 
of  A,  is  now  driven  to  a  greater  distance  ;  and  therefore  a  larger 
charge  will  come  from  the  remote  to  the  nearer  side  of  A,  which 
in  turn  attracts  more  negative  to  the  nearer  end  of  B  (7,  and  thus 
a  new  series  of  actions  and  reactions  takes  place  in  addition  to  the 
former.     To  obtain  the  greatest  effect  from  this  cause,  C  is  con- 
nected with  the  earth ;  then  the  positive  electricity  is  driven  to 
the  earth,  and  entirely  disappears,  and  the  negative  is  attracted  to 
the  nearer  end.     This  experiment  is  performed  by  touching  the 


DISGUISED    ELECTRICITY.  403 

finger  to  the  conductor,  after  it  has  become  electrified  by  induc- 
tion. The  electroscope  nearest  to  A  instantly  rises  a  little  higher, 
and  the  distant  ones  collapse. 

639.  Disguised  Electricity.— The  electricity  which  occu- 
pies the  surface  of  the  prime  conductor,  or  any  other  body  electri- 
fied 1:1  the  ordinary  way,  and  which  is  kept  from  diffusing  itself 
in  every  direction  only  by  the  pressure  of  the  air  (Art.  632),  is 
called  free  electricity  ;  for  it  will  instantly  spread  over  the  surface 
of  other  conductors,  when  it  touches  them,  and  therefore  will  be 
lost  in  the  earth,  the  moment  a  communication  is  made.    But  the 
electricity  which  is  accumulated  by  the  inductive  influence  is  not 
free  to  diffuse  itself;  the  same  attractive  force  which  has  con- 
densed it  still  holds  it  as  near  as  possible  to  the  original  charge  ; 
and  if  we  touch  the  electrified  body  with  the  hand,  the  electricity 
does  not  pass  off;  it  is  therefore  called  disguised  electricity.     In 
this  respect  the  two  fluids  on  the  contiguous  sides  of  A  and  B  C 
are  alike;  either  may  be  touched,  or  in  any  way  connected  with 
the  earth,  but,  unless  communication  is  made  between  them,  or 
unless  they  are  both  allowed  to  pass  to  the  earth,  they  hold  each 
other  in  place  by  their  mutual  attraction,  and  show  none  of  the 
phenomena  of  free  electricity. 

640.  A  Series  of  Conductors.— If  another  insulated  con- 
ductor, /),  is  placed  near  to  the  remote  end  of  B  C  (Art.  634),  and 
A  is  charged  positively,  then  that  extremity  of  B  C' nearest  to  D 
is  inductively  charged  with  positive,  as  already  stated.     Hence, 
the  electricities  of  D  are  separated,  the  negative  approaching  B  C, 
and  the  positive  withdrawing  from  it ;  there  is  therefore  the  same 
arrangement  of  fluids  in  both  bodies,  but  a  less  intensity  in  D 
than  in  B  C.     For,  on  account  of  distance,  the  positive  is  not  so 
intensely  accumulated  at  the  remote  end  of  B  C  as  in  the  original 
body  A,  and  therefore  a  less  force  operates  on  D  than  on  B  C 
The  same  effects  are  produced  in  a  less  and  less  degree  in  an 
indefinite  series  of  bodies ;  and  the  shorter  they  are,  the  more 
nearly  equal  will  be  the  successive  accumulations.      The  same 
facts  were  noticed  in  a  series  of  magnets. 

641.  The  Gold-Leaf  Electroscope.— The  gold-leaf  elec- 
troscope consists  of  two  narrow  strips  of  gold-leaf,  n,  n  (Fig.  355), 
suspended  within  a  glass  receiver,  B,  from  a  metallic  rod  which 
passes  through  the  top  and  terminates  in  a  ball,  C.    A  metallic 
base  is  cemented  to  the  receiver,  and  strips  of  tin-foil,  a,  are 
attached  to  the  inside,  reaching  to  the  base,  in  order  to  discharge 
the  gold  leaves  whenever  they  are  caused  to  diverge  excessively. 
Let  a  body  positively  electrified  be  brought  within  a  few  feet  of 
the  knob.      It  attracts   the  negative  from  the  leaves  into  the 


404      PRICTIONAL  OR  STATICAL  ELECTRICITY. 


knob,  and  repels  the  positive  from  the  knob  into  the  leaves; 
they  are  thus  electrified  alike,  and  repel  each  other.  If  the 
charged  body  is  brought  so  near  that  the 
leaves  touch  the  conductors,  which  are 
placed  on  the  sides  of  the  cylinder,  and  dis- 
charge their  induced  electricity  to  them, 
then  they  collapse.  After  this,  they  will 
diverge  again,  whether  the  electrified  body 
is  brought  still  nearer,  or  withdrawn;  if 
brought  nearer,  they  diverge  by  means  of  a 
new  portion  of  positive,  repelled  from  the 
knob ;  if  withdrawn,  they  diverge  by  the 
return  of  negative  electricity  from  the 
knob,  which  is  no  longer  neutralized  by  the 
positive,  since  the  latter  has  been  discharged 
to  the  earth. 

As  there  is  danger  of  rupturing  the  gold 
leaves  by  too  violent  action,  the  method  usually  adopted  is  as 
follows :  Let  a  charged  body,  positively  charged  for  example,  be 
brought  near  the  knob,  and  the  leaves  will  diverge,  being  charged 
positively  by  induction  ;  if  the  knob  be  now  touched  by  the 
finger,  the  inducing  body  remaining  as  Before,  the  leaves  will  at 
once  collapse,  the  repelled  positive  electricity  being  driven  to  earth. 
If  now  the  finger  be  removed,  and  afterward  the  inducing  body 
be  withdrawn,  the  negative  charge  will  diffuse,  causing  the  leaves 
to  diverge  by  mutual  repulsion. 

If  now  a  charged  body,  suspended  by  an  insulating  silk  thread, 
in  order  that  the  hand  may  be  kept  so  far  from  the  knob  as  to 
prevent  its  inductive  influence,  be  brought  near  the  knob,  an  in- 
crease of  divergence  would  indicate  a  negative  charge  upon  the 
body,  while  decrease  of  divergence  would  indicate  a  positive 
charge. 

642.  Mutual  Attractions  and  Repulsions  of  Bodies. — 

That  an  electrified  body  attracts  an  unelectrified  body,   a  fact 
which  is  among  the  first  to  be  noticed  in 
observing  electrical  phenomena  (Art.  621), 
is  explained  by  induction. 

1.  Let  A  (Fig.  356)  be  charged  positively, 
and  let  B  be  unelectrified.  A  acting  by  in- 
duction will  decompose  the  electricity  of 
B,  as  shown  in  Fig.  356,  attracting  the 
negative  to  the  nearest  point  c  and  repelling 
the  positive  to  the  furthest  point  d.  As  the 
distance  a  c  is  less  than  a  d,  then,  according 


FIG.  356. 


INDUCTIVE    ACTION    INCREASED. 


405 


to  the  law  given  in  Art.  623,  the  mutual  attraction  must  exceed 
the  repulsion,  and  the  bodies  will  move  toward  each  other. 

2.  Let  B  be  a  conductor  charged  negatively,  then  there  will  be 
mutual  attraction. 

3.  Let  B  be  a  conductor  charged  positively,  then  there  may  be 
repulsion  or  attraction  according  to  the  distance  a  c.     In  addition 
to  the  positive  charge  on  B  there  is  also  some  decomposed  electri- 
city.    This,  acted  upon  by  A,  is  separated,  the  positive  increasing 
the  repelled  positive  charge  at  d,  and  the  negative  appearing  at  c. 
At  a  distance  from  A,  B  will  be  repelled  ;  but  if  a  c  be  diminished 
the  attractive  force  increases  more  rapidly  than  the  repellent,  and 
finally  a  c  may  become  so  small  that  the  attraction  may  exceed  the 
repulsion. 

4.  If  B  be  a  poor  conductor  with  negative  charge,  attraction 
will  result ;  if  with  positive  charge,  repulsion  ensues.     If  it  be 
unelectrified  there  will  be  some  inductive  decomposition  of  its 
electricity,  and  attraction  will  be  shown. 

That  the  charge  does  not  leave  either  body,  but  constrains  each 
to  move  with  it,  is  explained  by  Art.  632. 

643.  The  Inductive  Action  Greatly  Increased. — In  the 

experiments  as  now  described,  the  inductive  influence  is  feeble, 
and  the  accumulation  of  electricities  very  small ;  for  the  bodies 
present  toward  each  other  only  a  limited  extent  of  area,  and  they 
are  necessarily  as  much  as  four  or  five  inches  distant,  in  order  to 
prevent  the  fluid  from  passing  across.  By  giving  the  bodies  such 
a  form  that  a  large  extent  of  surface  may  be  equidistant,  and  then 
interposing  a  solid  non-conductor,  as  glass,  between  them,  so  that 
the  distance  may  be  reduced  to  one-eighth  of  an  inch  or  less,  it  is 
easy  to  increase  the  attracting  and 
repelling  forces  many  thousands  of 
times  (Art.  629).  Let  a  glass  plate, 
C  D  (Fig.  357),  supported  on  a  base, 
have  attached  to  the  middle  of  each 
side  a  rectangular  piece  of  tin-foil. 
This  is  called  a  Franklin  plate.  Let 
A  be  connected  with  one  coating,  and 
B  with  the  other.  If,  now,  A  forms 
a  part  of  the  prime  conductor  of  an 
electrical  machine,  and  B  has  com- 
munication with  the  earth,  there  will 
he  accumulation  of  great  quantities 
of  electricity,  positive  on  one  side 

and  negative  on  the  other.     If  the  amount  of  surface  and  the 
thickness  of  glass  are  the  same,  the  particular  form  of  the  instru- 


FIG.  357. 


406      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

ment  is  immaterial ;  but,  for  most  purposes,  a  vessel  or  jar  is 
more  convenient  than  a  pane  of  glass  of  equal  surface,  and  is 
generally  employed  for  electrical  experiments. 

644.  The  Condensing  Electroscope. — Instruments  called 
by  this  name  are  intended  for  the  accumulation  of  electricity  from 
some  feeble  source,  until  it  may  be  rendered  sensible.     The  most 
delicate  is  the  gold-leaf  condenser.    Suppose  the  gold-leaf  electro- 
scope to  have  a  disk,  A,  instead  of  a  knob  on  the  top  (Fig.  358). 

Another  disk,  B,  is  furnished  with  an  insulating  handle, 
Pig.  358.  an(j  De£ween  the  disks  is  placed  the  thinnest  possible 
non-conductor,  as  a  film  of  varnish.  Set  the  plate 
B  upon  A,  and  place  the  finger  upon  the  upper  surface 
of  B.  Then  touch  the  under  surface  of  A  with  the 
body  feebly  charged ;  the  charge  will  diffuse  itself  over 
the  upper  surface  of  A  and  will  induce  an  opposite 
charge  on  the  near  surface  of  B.  Apply  successive 
weak  charges  in  like  manner,  all  of  which  will  accumu- 
late upon  the  upper  surface  of  A.  Thus  far  the  gold 
leaves  have  shown  no  indications  of  electricity,  the 
charge  upon  A  being  condensed  upon  its  surface  by  the 
opposite  induced  charge  in  B ;  but  on  taking  up  B  by  the  insulat- 
ing handle,  the  electricity  condensed  in  A  is  set  free,  flows  down  to 
the  leaves  and  repels  them,  thus  rendering  the  accumulation  per- 
ceptible. 

645.  Influence  of  the  Interposed  Non-Conductor. — If 

instead  of  a  glass  plate  between  the  two  metallic  surfaces  in  the 
Franklin  plate,  a  film  of  air  of  the  same  thickness  be  substituted, 
the  inductive  action  will  be  about  one-third  as  great  as  before, 
and  if  a  plate  of  sulphur  had  been  used  instead  of  glass  the  effect 
would  have  been  five-sixths  as  great  as  with  glass.  The  effect  of 
the  interposed  insulator,  or  dielectric,  may  be  shown  thus  :  Let 
the  charged  disk  B  (Fig.  358)  be  placed  above  A  at  a  distance 
which  will  cause  the  leaves  to  diverge  slightly,  as  in  the  figure. 
If  now  a  plate  of  any  dielectric,  as  rubber  or  paraffine,  be  inter- 
posed between  the  disks,  the  gold  leaves  will  separate  more  widely, 
showing  the  inductive  influence  of  the  dielectric.  The  following 
table  is  taken  from  results  given  by  Gordon : 


Specific  Inductive  Capacities. 

Air  or  any  gas 1 . 00 

Paraffine...  .1.99 


Specific  Inductive  Capacities. 

Gutta-percha 2.46 

Sulphur 2. 58 


Black  India-rubber 2.22        Shellac 2.74 

Ebonite....": ...  2.28    |    Glass 3.25 

646.  The  Leyden  Jar.— This  article  of  electrical  appara ins 


THEORY  OF  THE    LEYDEN  JAR.  407 

consists  of  a  glass  jar  (Fig.  359),  coated  on  both  sides  with  tin-foil, 
except  a  breadth  of  two  or  three  inches  near  the  top,  which  is 
sometimes  varnished  for  more  perfect  insulation.  Through  the 
cork  passes  a  brass  rod,  which  is  in  metallic  contact  with  the 
inner  coating,  and  terminates  in  a  ball  at  the  top. 

On  presenting  the  knob  of  the  jar  near  to  the  prime  conductor 

FIG.  359. 

FIG.  360. 


of  an  electrical  machine,  while  the  latter  is  in  operation,  a  series 
of  sparks  passes  between  the  conductor  and  the  jar,  which  will 
gradually  grow  more  and  more  feeble,  until  they  cease  altogether. 
The  jar  is  then  said  to  be  charged.  If  now  we  take  the  discharg- 
ing-rod,  which  is  a  curved  wire,  terminated  at  each  end  with  a 
knob,  and  insulated  by  glass  handles  (Fig.  360),  and  apply  one  of 
the  knobs  to  the  outer  coating  of  the  jar,  and  bring  the  other  to 
the  knob  of  the  jar,  a  flash  of  intense  brightness,  accompanied  by 
a  sharp  report,  immediately  ensues.  This  is  the  discharge  of  the 
jar. 

If,  instead  of  the  discharging-rod,  a  person  applies  one  hand 
to  the  outside  of  the  charged  jar,  and  brings  the  other  to  the 
knob,  a  sudden  shock  is  felt,  convulsing  the  arms,  and  when  the 
charge  is  heavy,  causing  pain  through  the  body.  The  shock  pro- 
duced by  electricity  was  first  discovered  accidentally  by  persons 
experimenting  with  a  charged  phial  of  water.  This  occurred  in 
Leyden,  and  led  to  the  construction  and  name  of  the  Leyden  jar. 

647.  Theory  of  the  Leyden  Jar. — This  instrument  accu- 
mulates and  condenses  great  quantities  of  electricity  on  its  sur- 
faces, upon  the  principle  of  mutual  attraction  between  unlike 
electricities,  one  of  which  is  furnished  by  the  machine,  the  other 
obtained  from  the  earth  by  induction.  First,  suppose  the  outer 
coating  insulated  ;  a  spark  of  the  positive  electricity  passes  from 
the  prime  conductor  to  the  inner  coating,  which  tends  to  repel 
the  positive  from  the  outer  coating  ;  but  as  the  latter  cannot 
escape,  it  remains  to  prevent,  by  its  counter-repulsion,  any  addi- 


408      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

tion  to  the  charge  of  the  inside,  and  thus  the  process  stops.  But 
now  connect  the  outer  coating  with  the  earth,  and  immediately 
some  of  its  positive  electricity,  repelled  by  the  charge  on  the 
inside,  passes  off,  while  its  negative  is  attracted  close  upon  the 
glass.  The  negative  upon  the  outside,  by  its  attraction,  con- 
denses the  positive  of  the  inner  coating,  and  a  second  spark  passes 
in  from  the  prime  conductor.  This  produces  the  same  effect 
as  the  first,  and  a  second  addition  of  negative  is  made  to  the 
outer  coating,  the  latter  being  obtained  from  the  earth  as  before. 
These  actions  and  reactions  go  on  in  a  diminishing  series,  till  there 
is  a  great  accumulation  of  the  two  electricities,  held  by  mutual 
attraction  as  near  each  other  as  possible,  on  opposite  sides  of  the 
glass.  The  jar  in  this  condition  is  said  to  be  charged. 

If  the  positive  electricity  is  on  the  inner  coating,  the  jar  is 
said  to  be  positively  charged;  if  on  the  outside,  negatively 
charged. 

648.  The  Spontaneous  Discharge.— This  occurs  when  the 
quantities  accumulated  are  so  great  that  their  attraction  will  cause 
them  to  fly  together  with  a  flash  and  report  over  the  edge  of  the 
jar.     If  the  glass  is  soiled  or  damp,  the  fluids  may  pass  over  and 
mingle  with  only  a  hissing  noise,  in  which  case  it  is  impossible  for 
the  jar  to  be  highly  charged. 

If  the  glass  is  clean  and  dry,  and  especially  if  varnished  with 
gum  lac,  a  charge  may  not  wholly  disappear  for  days,  or  even 
weeks. 

649.  Series  of  Jars. — The  same  amount  of  electricity  from 
the  prime  conductor  which  is  required  to  charge  one  jar  will  charge 
an  indefinite  series,  the  strength  of  the  charge  being  less  and  less 
from  the  first  to  the  last.     This  case  is  analogous  to  the  series  of 
conductors  (Art.  640).     Insulate  a  series  of  jars,  A,  B,  C,  &c.,  and 
connect  the  inner  coating  of  A  with  the  prime  conductor,  and  its 
outer  coating  with  the  inner  coating  of  B,  the  outer  of  B  with 
the  inner  of  C,  and  so  on.     Then,  as  A  is  charged,  the  positive 
electricity  of  its  outer  coating,  instead  of  passing  to  the  earth, 
goes  to  the  inside  of  B,  and  that  on  the  outside  of  B  to  the  inside 
of  (7,  &c.,  while  that  on  the  outside  of  the  last  in  the  series  passes 
to  the  earth.    Thus  each  jar  is  charged  positively  by  the  inductive 
influence  of  the  preceding,  just  as  a  series  of  magnets  is  formed 
with  poles  in  the  same  order  by  a  succession  of  magnetic  induc- 
tions.   No  gain  results  from  this  arrangement  however,  for  the 
total  charge,  or  sum  of  the  separate  charges,  is  only  equal  to  that 
which  a  single  jar  would  have  taken  under  the  given  conditions. 

650.  Division  of  a  Charge  in  any  Given  Ratio. — If  one 


USE    OF    THE    COATINGS.  409 

of  two  jars  be  charged,  and  the  other  not,  and  if  the  inner  coat- 
ings be  brought  into  communication,  and  also  the  outer  coatings, 
the  charge  of  the  first  jar  is  instantly  diffused  over  the  two,  with 
a  report  like  that  of  a  discharge.  In  this  way  a  charge  may  be 
halved,  or  divided  in  any  other  ratio,  according  to  the  relative 
capacities  of  the  jars. 

The  capacity  of  a  jar  with  equal  coatings  is  proportional  to  the 
area  of  either  coating,  inversely  proportional  to  the  thickness  of 
the  dielectric,  and  proportional  to  the  specific  inductive  capacity 
of  the  dielectric.  Thus  two  jars  of  the  same  kind  of  glass,  of 
equal  thickness  and  having  equal  coatings,  will  be  of  equal 
capacity. 

The  self-repellency  of  each  charge  tends  to  diffuse  it  over  a 
greater  surface,  and  they  will  be  thus  diffused  if  allowed  to  remain 
within  each  other's  attracting  influence ;  but  one  of  the  charges 
will  not  spread  over  the  coatings  of  another  jar,  unless  oppor- 
tunity is  given  for  both  to  do  so. 

An  experiment  somewhat  resembling  the  foregoing  is  this  : 
charge  two  equal  jars,  one  positively,  the  other  negatively,  and 
insulate  them  both.  If  the  two  knobs  be  connected  by  a  conduc- 
tor, the  electricities,  notwithstanding  their  strong  attraction,  will 
not  unite;  for  each  is  held  disguised  by  that  on  the  other  side  of 
the  glass.  But  if  the  outer  coatings  are  first  connected,  then, 
on  joining  the  knobs,  the  jars  are  both  discharged  at  once. 

651.  Use  of  the  Coatings. — If  a  jar  is  made  with  a  wide 
open  top,  and  the  coatings  movable,  then,  after  charging  the  jar 
and  removing  the  coatings,  very  little  of  the  electricities  adheres  to 
the  latter,  but  nearly  the  whole  remains  on  the  glass.     The  same 
mutual  attraction  which  condensed  them  at  first  still  holds  them 
there  after  the  coatings  are  removed.     When  they  come  to  be  re- 
placed, the  jar  can  be  discharged  as  usual.     But  the  coatings  are 
necessary  in  charging,  to  diffuse  the  electricity  over  those  parts  of 
the  glass  which  they  cover,  and  also  in  discharging,  to  conduct  off 
the  whole  charge  at  once. 

652.  The  Free   Portion  of  an  Electrical  Charge. — 

Either  kind  of  electricity  is  said  to  be  free  when  it  remains  on  a 
body  only  because  held  by  the  pressure  of  the  air  ;  but  if  held  by 
the  attraction  of  the  opposite  kind,  it  is  said  to  be  disguised  (Art. 
639).  Nearly  all  the  electricity  of  a  charged  jar  is  disguised,  but 
not  the  whole. 

The  moment  after  a  jar  is  charged  there  is  a  small  quantity  of 
free  electricity  on  the  coating  to  which  the  fluid  was  furnished  in 
charging,  but  not  on  the  other.  If  the  charged  jar  be  upon  an 
insulating  stand,  and  the  finger  brought  to  this  coating,  a  slight 


410      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

spark  is  taken  off ;  if  it  be  touched  again  immediately,  there  is 
no  spark,  for  the  free  electricity  all  escaped  by  the  first  contact. 
Let  the  finger  now  be  brought  to  the  other  coating,  and  a  spark 
flies  from  that.  Immediately  afterward  a  second  spark  can  be 
taken  from  the  first  coating,  and  so  on  alternately  for  hundreds 
of  times  usually  before  the  charge  wholly  disappears.  What  is 
removed  at  each  contact  is  the  free  part  of  the  charge,  which 
always  appears  alternately  on  the  two  coatings.  If  a  small  electro- 
scope be  connected  with  each  coating,  the  fluid  alternately  set  free 
is  indicated  to  the  sight.  The  electroscope  on  the  coating  which 
is  touched  instantly  falls,  and  the  other  rises. 

653.  Explanation  of  this  Phenomenon. — The  positive 
electricity  which  is  conveyed  to  the  inner  coating,  in  charging  a 
jar,  attracts  to  the  outer  coating  from  the  earth  a  quantity  of  the 
negative  fluid  which  is  a  little  less  than  itself.     This  is  because  of 
the  thickness  of  the  glass.     If  it  were  infinitely  thin,  the  negative 
would  be  just  equal  to  the  positive,  and  they  would  neutralize  each 
other,  and  both  be  perfectly  disguised.    But  as  the  glass  has  some 
thickness,  the  positive  exceeds  the  negative,  and  disguises  it.    Now 
if  the  jar,  after  being  charged,  is  insulated,  it  is  obvious  that  the 
negative  charge  on  the  outer  coating  cannot  disguise  all  the  posi- 
tive (which  is  more  than  itself),  but  only  a  quantity  a  little  less 
than  itself.     Hence  there  must  be  a  little  of  the  positive  on  the 
inner  coating  in  a  free  state.     By  touching  the  knob,  we  allow 
this  free  portion  to  pass  off,  and  there  is  left  less  of  the  positive 
in  the  inner  coating  than  there  is  of  the  negative  in  the  outer. 
Therefore,  all  the  negative  cannot  now  be  disguised,  but  a  slight 
quantity  is  liberated  and  ready  to  pass  off  as  soon  as  touched.   And 
thus,  by  alternate  contacts,  the  process  of  discharge  goes  on,  the 
series  being  longer  as  the  glass  is  thinner,  because  then  the  two 
charges  are  more  nearly  equal,  and  less  electricity  is  set  free  after 
the  alternate  contacts. 

654.  Electrical   Vibrations. — If  two  jars  be  charged  in 
opposite  ways,  and  a  figure  made  of  pith  be  suspended  between  the 
knobs  by  a  long  thread,  it  will  be  attracted  by  that  knob  whose 
action  on  it  happens  to  be  greatest.     As  soon  as  it  touches,  it  is 
charged  with  electricity  and  repelled,  and  of  course  attracted  by 
the  other  knob,  which  is  in  the  opposite  state  ;  thus  it  vibrates 
between  them,  causing  a  very  slow  discharge  of  both  jars.     In  this 
case,  the  outside  of  each  jar,  which  must  not  be  insulated,  dis- 
charges the  free  electricity,  resulting  from  the  contact  of  the 
figure  with  the  knob  of  its  inner  coating,  directly  to  earth,  thus 
setting  free  upon  its  knob  a  new  portion  of  the  inner  charge  as 
explained  in  Art.  653. 


RESIDUARY    CHARGE.  411 

The  electricity  of  the  prime  conductor  will  also  cause  vibra- 
tions, without  the  use  of  a  jar.  Suspend  from  it  a  metallic  disk 
horizontally  a  few  inches  above  another  which  is  connected  with 
the  earth  ;  then  if  a  glass  cylinder  surround  the  two  disks  so  as  to 
prevent  escape,  a  number  of  pith  balls  between  the  disks  will  con- 
tinue to  vibrate  up  and  down  so  long  as  the  machine  is  in  action. 
Each  ball  lying  on  the  lower  disk,  being  electrified  by  induction, 
springs  up  to  the  upper  disk,  and  then,  being  charged  in  the 
same  way,  is  repelled. 

In  a  similar  manner  a  chime  of  bells  may  be  rung. 

655.  Residuary   Charge. — If  a  jar  stand  charged  a  few 
minutes,  and  after  the  discharge  remain  some  minutes  more,  then 
a  second,  and  possibly  a  third,  discharge  can  be  made ;  but  these 
are  usually  very  slight.     The  electricity  remaining  after  the  first 
discharge  is  called  the  residuary  charge.     The  larger  the  jar,  and 
the  greater  the  density  of  the  charge,  the  larger  is  this  residuum. 
The  outer  and  inner  charges  not  only  attract  each  other  to  the 
surfaces  of  the  intervening  dielectric,  but  also  penetrate  its  sub- 
stance sensibly.     When  the  jar  is  discharged  the  portion  of  the 
charge  thus  absorbed  can  not  pass  off  instantly  because  of  the 
poor  conductivity  of  the  dielectric.     After  some  moments  a  new 
distribution  of  the  residual  electricity  has  taken  place,  much  of 
it  having  returned  to  the  surface  of  the  coatings,  and  a  second 
discharge  is  possible.     It  is  probable  that  there  is  also  some  diffu- 
sion of  the  charge  above  the  edges  of  the  coatings,  the  gradual 
return  of  which  adds  to  the  intensity  of  the  residual  charge. 

656.  The   Electric   Battery. — Leyden  jars  are  made  of 
various  sizes,  from  a  half-pint  to  one  or  two  gallons.     But  when 
a  great  amount  of  surface  is  needed,  it  is  more  convenient,  and,  in 
case  of  fracture  by  violent  discharge,  more  economical,  to  connect 
several  jars,  so  that  they  may  be  used  as  one.    Four,  nine,  twelve, 

or  even  a  greater  number  of  jars,  are 
set  in  a  box  (Fig.  361),  whose  inte- 
rior is  lined  with  tinfoil,  so  as  to  con- 
nect all  the  outer  coatings,  together. 
Their  inner  coatings  are  also  con- 
nected, by  wires  joining  all  the 
knobs,  or  by  a  chain  passing  round 
all  the  stems.  Care  is  necessary  in 
discharging  batteries,  that  the  cir- 
cuit is  not  too  short  and  too  perfect, 

since  the  violence  of  discharge  is  liable  to  perforate  the  jars.  A 
chain,  three  or  four  feet  long  in  the  circuit,  will  generally  prevent 
the  accident. 


FUICTIONAL   OR    STATICAL    ELECTRICITY. 


FIG.  362. 


657.  Discharging  Electrometers.— These  are  instruments 
contrived  for  measuring  the  charge  in  the  act  of  discharging  the 
jar.  Fig.  362  represents  Lane's  discharging  electrometer.  D  is 
a  rod  of  solid  glass,  which  holds  a  sliding 
metallic  rod  with  balls  B  and  C. 

If  the  circuit  through  which  the  charge 
is  to  be  sent  is  made  to  begin  at  the  knob  0, 
and  end  with  the  outer  coating  of  the  elec- 
trometer, and  if  the  knob  A  be  connected 
with  the  prime  conductor  of  a  machine,  the 
inner  coating  of  the  electrometer  will  acquire 
a  high  potential  which  will  cause  a  discharge 
across  the  intervening  air  between  A  and  B. 
The  distance  between  A  and  B  may  be  made 
great  or  small  at  will,  and  hence  a  succession  of  charges  of  greater 
or  less  intensity  may  be  caused  to  traverse  the  circuit. 

The  unit  jar  is  used  to  measure  the  charge  of  another  jar,  by 
conveying  to  it  successive  equal  charges  of  its  own.  A  B  (Fig. 
363)  is  the  instrument,  consisting  of  a  small  open  jar,  placed 

FIG.  363. 


horizontally  on  an  insulating  stand,  B.  From  the  metallic  part 
of  the  support,  the  bent  rod  and  ball,  C,  come  near  to  D,  the  rod 
of  the  inner  coating,  and  can  be  turned  so  as  to  increase  or 
diminish  its  distance.  Let  the  knob,  D,  be  near  the  prime  con- 
ductor, and  that  of  the  outer  coating  near  the  top  of  the  jar,  E, 
which  is  to  be  charged,  the  outside  of  the  latter  being  in  com- 
munication with  the  earth.  While  A  is  charging,  the  positive 
electricity  of  its  outer  coating  goes  to  the  inner  coating  of  the 
large  jar,  and  partially  charges  it,  the  negative  being  held  by 
induction  upon  the  outer  coating  of  A.  When  the  difference 
of  potential  is  sufficiently  great,  a  discharge  takes  place  between 
D  and  C,  and  the  outer  and  inner  coatings  of  A  become  neutral 
again.  A  second  charge  of  positive  is  now  repelled  into  the 
jar  E,  by  the  continued  working  of  the  electric  machine,  as 
before,  and  so  on  till  the  required  number  of  unit  charges  have 
passed. 


THE    ELECTROPHORUS.  413 

658.  The  Effect  of  a  Point  presented  to  an  Electri- 
fied Body. — It  has  been  noticed  (Art.  631)  that  a  pointed  wire 
wastes  the  charge  very  quickly,  because  of  the  accumulation  at 
the  point.     A  prime  conductor  loses  its  charge  just  as  quickly  by 
presenting  a  pointed  rod  toward  it.     For  the  induced  electricity 
of  the  rod  and  person  holding  it  is  in  like  manner  accumulated 
at  the  point,  arid  readily  escapes  to  mingle  with  and  neutralize  its 
opposite  in  the  prime  conductor.     Thus  the  charge  disappears  at 
once.     In  a  similar  way  is  to  be  explained  the  use  of  the  points  on 
the  prime  conductor  presented  to  the  glass  plate.     When  the  two 
electricities  are  separated  at  the  surface  of  contact  between  the 
plate  and  rubbers,  the  plate  is  positively  electrified.     This  positive 
charge  acts  inductively  on  the  prime  conductor,  attracting  the 
negative  kind  to  the  points,  where  it  passes  off  and  neutralizes 
what  is  on  the  plate,  and  leaves  a  positive  charge  on  the  prime 
conductor,  thus  charging  it,  not  by  giving  to  it  positive  electricity, 
but  by  robbing  it  of  its  negative,  and  thus  leaving  its  positive 
upon  it. 

659.  The  Electrophorus. — This  is  a  very  simple  electrical 
machine  for  giving  the  spark.     It  consists  of  a  circular  cake  of 
resin  in  a  metallic  base,  A  (Fig.  364),  and  a  metallic  disk,  B, 
having  a  glass  handle.     Excite  the  resin  by  fur  or  flannel ;  set  the 
disk  B  upon  it,  and  touch  the  back  of  the 

disk  with  the  finger.  Having  withdrawn 
the  finger  again,  lift  the  disk  B  by  its 
insulating  handle,  and  it  will  be  found 
charged  positively,  and  a  brilliant  spark 
may  be  drawn  from  it.  Again  place  the 
disk  upon  the  resin,  touch  it  with  the 
finger  as  before,  and  then  remove  it,  and  a 
second  charge  may  be  taken  from  it,  and 
so  on  indefinitely. 

The  negative  electricity  of  the  resin,  the  result  of  the  friction 
by  the  fur,  does  not  reside  wholly  upon  the  surface  (Art.  655),  but 
penetrates  the  dielectric.  This,  by  induction,  produces  a  positive 
charge  in  the  metallic  plate  A,  and  the  two  charges  are  held  in 
place  by  their  mutual  attractions.  When  B  is  set  upon  A  a  posi- 
tive charge  is  drawn  to  its  under  surface  by  the  inductive  action 
of  the  negative  charge,  while  the  negative  of  B  is  repelled  to 
earth  through  the  finger.  Upon  removing  the  finger  and  lifting 
the  plate  B,  the  positive  charge,  no  longer  disguised,  rises  in 
potential  (Art.  629),  and  may  be  drawn  off  in  a  spark. 

660.  The  Dielectric  Machine. — In  this  machine  the  lower 


414      FRICTIONAL  OR  STATICAL  ELECTRICITY. 


FIG.  365. 


revolving  vulcanite  disk  A  (Fig.  365)  is  excited  by  the  clamped 

rubbers  at  By  as  it  is  made  to  revolve  as  indicated  by  the  arrow. 

The  upper  plate  (7,  also  of 
vulcanite,  is  caused  to  re- 
volve in  the  opposite  direc- 
tion, in  close  proximity  to 
the  combs  at  D  and  E. 
The  supports  of  the  prime 
conductor  H  are  of  glass  or 
other  insulating  material. 

Suppose  that  negative 
electricity  is  developed  upon 
the  disk  A  by  the  friction 
of  the  rubbers  at  B  ;  as  this 
charge  passes  behind  the 
revolving  disk  C  it  induces 
a  positive  charge  in  the 
combD,  which  is  discharged 
upon  the  front  of  C,  the 
negative  of  D  and  its 
connections  being  repelled 

through  the  knob  and  rod  to  the  knob  near  the  prime  conductor 

H.    The  plate  C  carries  the  positive  charge  it  has  received  around 

to  the  comb  E,  inducing  in 

it  negative  electricity,  which 

is  discharged  upon  the  plate, 

restoring  it  to  its    neutral 

state,  while  the  positive  of 

E  is  repelled  into  the  prime 

conductor  H. 

Sparks    from    seven    to 

twelve  inches  long  may  be 

made  to   pass  between  the 

conductor  #and  the  knob  of 

the  adjustable  rod. 

661.  The  Holtz  Ma- 
chine,— This  machine, being 
named  from  its  inventor  and 
illustrated  in  Fig.  366,  con- 
sists of  a  revolving  glass 
disk  A  and  a  stationary  glass 
disk  B,  both  well  coated  with 
shellac  to  guard  against  mois- 
ture. In  front  of  A  and  close 


FIG  ?66. 


GENERAL    EXPLANATION. 


415 


to  it,  as  shown  in  the  figure,  are  two  combs,  as  in  the  dielectric 
machine,  connected  with  the  discharging  knobs  at  C.  The  inner 
coatings  of  two  Leyden  jars  D  are  connected  with  their  respective 
knobs  and  combs,  the  outer  coatings  of  the  jars  being  connected 
together.  On  the  back  of  the  disk  J5,  opposite  to  the  combs,  are 
two  paper  sectors,  a  paper  tongue  or  point  from  each  projecting, 
through  an  opening  in  the  stationary  disk,  towards  the  revolving 
plate.  If  a  plate  of  vulcanite  be  excited,  and  then  be  laid  against 
one  of  the  paper  sectors,  while  the  disk  A  is  rapidly  rotated,  the 
discharging  knobs  being  in  contact,  electrical  decomposition  will 
ensue,  and  after  a  few  moments  the  knobs  may  be  gradually 
separated  until  they  give  sparks  twelve  or  twenty  inches  long, 
according  to  the  size  of  the  machine. 

To  explain,  in  a  very  general  way,  the  action  of  the  machine, 
let  A  (Fig.  367),  represent  the  revolving  plate  and  B  the  station- 
ary plate  behind  it,  carrying  the  paper 
sectors  a  and  b.  Imagine  the  combs  to 
be  in  front  of  A,  and  call  them  a  and  b 
also,  remembering  that  the  sectors  are 
on  the  plate  B,  behind  A.  If  now  a 
positive  charge  be  communicated  to  the 
sector  a  it  will  act  by  induction,  as  in 
the  dielectric  machine,  and  will  cause  the 
comb  a  to  discharge  negative  electricity 
upon  the  front  of  the  plate  A,  the  posi- 
tive of  the  comb  being  repelled.  This 
repelled  positive  charge  passes,  the  knobs 
being  in  contact,  to  the  comb  b,  and  in- 
duces a  negative  charge  in  the  sedor  b,  the  positive  electricity  of 
the  sector  b  passing  off  by  means  of  the  pointed  tongue  attached 
to  it.  As  the  plate  A  is  now  made  to  revolve  the  constant  dis- 
charge of  the  comb  a  electrifies  its  lower  half  negatively,  while  the 
discharge  of  the  comb  b,  under  the  influence  of  the  sector  b,  charges 
the  upper  half  of  the  plate  positively.  The  machine  differs  from 
the  preceding  in  the  application  of  some  device,  such  as  the 
pointed  slips  of  paper  attached  to  each  paper  sector,  by  which  the 
repelled  electricities  of  the  sectors  may  escape,  thus  strengthening 
their  inductive  action  until  a  great  difference  of  potential  is 
established  between  the  two  discharging  knobs. 

The  Leyden  jars  increase   the  capacity  of  the  discharging 
knobs. 


416      FRICTIONAL    OR    STATICAL    ELECTRICITY. 


CHAPTER    IV. 

EFFECTS  OF  ELECTRICAL  DISCHARGES. 

662.  Variety  of  Effects.— Some  of  the  effects  of  electrical 
discharges  have  been  incidentally  noticed  in  the  foregoing  chap- 
ters.    The  bright  light,  the  sharp  sound,  and  the  great  suddenness 
of  the  transmission,  are  remarkable  phenomena  in  every  discharge 
of  a  Leyden  jar  or  battery.     The  various  effects  may  be  classified 
as  luminous,  mechanical,  chemical,  magnetic,  and  physiological. 

663.  Luminous  Effects. — Light  is  seen  only  when  charges 
of  electricity,  differing  greatly  in  potential,  are  discharged  through 
an  obstructing  medium.     Hence,  no  light  is  perceived  when  it 
flows  through  a  good  conductor,  unless  of  very  small  diameter. 
But  if  there  is  the  least  interruption,  or  if  the  conductor  is  re- 
duced to  a  very  slender  form,  then  light  appears  at  the  interrup- 
tion, and  at  those  parts  which  are  too  small  to  convey  the  electri- 
city.    Thus,  the  discharge  of  a  battery  through  a  chain  gives  a 
brilliant  scintillation  at  every  point  of  contact  between  the  links. 

664.  Modifications  of  the  Light.— The  length,  color,  and 
form  of  the  electric  spark  vary  with  the  nature  and  form  of  the 
conductors  between  which  it  passes,  and  with  the  quality  of  the 
medium  interposed  between  them. 

Electrical  sparks  are  more  brilliant  in  proportion  as  the  sub- 
stances between  which  they  occur  are  better  conductors.  A  spark 
received  from  the  prime  conductor  upon  a  large  metallic  ball  is 
short,  straight,  and  white  ;  on  a  small  ball  it  is  longer,  and  crooked  ; 
received  on  the  knuckle,  a  less  perfect  conductor,  the  middle  part 
is  purplish  ;  on  wood,  ice,  a  wet  plant,  or  water,  it  is  red. 

From  a  point  positively  electrified,  the  electricity  passes  in  the 
form  of  a  faint  brush  or  pencil  of  rays  ;  a  point  connected  with 
the  negative  side  exhibits  a  luminous  star. 

When  electricity  passes  through  rarefied  air,  the  light  becomes 
faint,  and  is  generally  changed  in  color.  The  electrical  spark, 
which  in  common  air  is  interrupted,  narrow,  and  white,  becomes, 
as  the  rarefaction  proceeds,  continuous,  diffused,  and  of  a  violet 
color,  which  tint  it  retains  as  long  as  it  can  be  seen.  If  a  battery 
is  discharged  through  a  tube  several  feet  long,  nearly  exhausted 
of  air,  the  whole  space  is  filled  with  a  rich  purple  light.  The 


LEIGH TENBERG     FIGURES.  417 

sparks  from  the  machine,  conveyed  through  the  same  tube,  exhibit 
flashings  and  tints  exceedingly  resembling  the  Aurora  Borealis. 

The  Geissler  tubes  are  tubes  of  complex  forms,  and  containing 
a  slight  trace  of  some  gas  or  vapor,  which  show  various  colors 
and  intensities  of  electric  light,  according  to  the  kind  of  gas,  the 
diameter  of  the  parts,  and  the  quality  of  the  glass.  The  electri- 
city is  conveye'd  into  the  tubes  by  platinum  wires  sealed  into  their 
extremities. 

Various  colors  are  obtained  by  sending  charges  through  dif 
ferent  substances.     An  egg  is  bright  crimson  ;  the  pith  of  corn- 
stalk, orange  ;  fluor-spar,  green  ;  and  loaf-sugar,  white  and  phos- 
phorescent. 

If  the  spark  be  examined  by  means  of  the  electroscope  it  will 
be  found  that  the  spectrum  indicates  that  the  substance  of  the 
conductors  has  been  volatilized,  and  that  the  air  or  gas,  through 
which  the  discharge  has  taken  place,  is  also  incandescent,  which 
facts  explain  the  differences  of  color  noted  above. 

665.  The  Leichtenberg  Figures. — When  a  spark  of  elec- 
tricity is  laid  upon  a  non-conductor,  it  will,  by  its  own  self-repel- 
lency,  extend  itself  a  little  distance  along  the  surface.     The 
Leichtenberg  figures  furnish  a  visible  illustration  of  this  fact,  and 
also  show  that  the  two  fluids  diffuse  themselves  in  very  different 
forms.    Lay  down  sparks  of  positive  electricity  from  the  knob  of 
the  Leyden  jar  upon  a  plate  of  resin,  and  near  them  some  sparks 
of  negative  electricity.     Then  blow  upon  the  plate  the  mingled 
powders  of  sulphur  and  red  lead.    The  sulphur,  by  the  agitation 
of  passing  through  the  air,  will  be  electrified  negatively,  and 
attracted  therefore  by  the  positive  sparks ;  the  red-lead,  positively 
electrified,  will  be  attracted  by  the  negative.     Thus  the  spots  on 
which  the  electricities  are  placed  will  appear  in  their  exact  forms 
by  means  of  the  colored  powders  attached  to  them.     The  positive 
resemble  stars,  or  rather  a  group  of  crystals  shooting  out  from  a 
nucleus ;  the  negative  spots  are  circles  with  smooth  edges ;  and 
the  size  of  the  electrified  spots  in  each  case  depends  on  the 
quantity  of  electricity  in  the  spark. 

666.  Luminous  Figures. — Metallic  conductors,  if  of  suffi- 
cient size,  transmit  electricity  without  any  luminous  appearance, 
provided  they  are  perfectly  continuous  ;  but  if  they  are  separated 
in  the  slightest  degree,  a  spark  will  occur  at  every  separation.     On 
this  principle,  various  devices  are  formed,  by  pasting  a  narrow 
band  of  tinfoil  on  glass,  in  the  required  form,  and  cutting  it 
across  with  a  penknife,  where  we  wish  sparks  to  appear.    If  an 
interrupted  conductor  of  this  kind  be  pasted  round  a  glass  tube 
in  a  spiral  direction,  and  one  end  of  the  tube  be  held  in  the  hand, 

27 


418      FRICTIONAL  OR  STATICAL    ELECTRICITY. 


FIG.  368. 


and  the  other  be  presented  to  an  electrified  conductor,  a  coil  of 
brilliant  points  surrounds  the  tube.  Words,  flowers,  and  other 
complicated  forms,  are  also  produced  nearly  in  the  same  manner, 
by  a  suitable  arrangement  of  interruptions  in  a  narrow  line  of  tin- 
foil, running  back  and  forth  on  a  plate  of  glass. 

667.  Mechanical  Effects. — Powerful  electric  discharges 
through  imperfect  conductors  produce  certain  mechanical  effects, 
such  as  perforating,  tearing,  or  breaking  in  pieces,  which  are  all 
due  to  the  sudden  and  violent  repulsion  between  the  electrified 
particles. 

A  discharge  through  the  air  is  supposed  to  perforate  it.  If 
the  air  through  which  the  spark  is  passed  lies  partially  inclosed 
between  two  bodies  which  are  easily  moved,  the  force  by  which 
the  air  is  rent  will  drive  them  asunder.  Thus,  a  little  block  may 
be  driven  out  from  the  foundation  of  a 
miniature  building,  and  the  whole  be 
toppled  down.  But  this  enlargement  of 
inclosed  air  is  best  seen  in  Kinnersley's 
air  thermometer  (Fig.  368).  As  the  spark 
passes  between  the  knobs  in  the  large 
tube,  the  air  confined  in  it  is  suddenly 
driven  asunder,  so  as  to  press  the  water 
which  occupies  the  lower  part  two  or 
three  inches  up  the  tube,  as  represented. 
As  soon  as  the  discharge  has  occurred,  the 
water  quietly  returns  to  its  level.  The 
sharp  sound  which  is  produced  by  the  dis- 
charge of  a  Leyderi  jar  is  due  to  the  sud- 
den compression  of  the  air,  and  also  to  the 
collapse  which  immediately  succeeds. 

The  path  of  the  electric  spark  through 
the  air,  when  short,  is  straight ;  but  if 
more  than  about  four  inches  long,  is  usually  entitled.  This  is 
supposed  to  arise  from  the  condensation  of  the  air  before  it,  by 
which  it  is  continually  turned  aside. 

When  the  charge  is  passed  through  a  thick  card,  or  the  cover 
of  a  book,  a  hole  is  torn  through  it,  which  presents  the  rough 
appearance  of  a  burr  on  each  side.  By  means  of  the  battery,  a 
quire  of  strong  paper  may  be  perforated  in  the  same  manner;  and 
such  is  the  velocity  with  which  the  fluid  moves,  that  if  the  paper 
be  freely  suspended,  not  the  least  motion  is  communicated  to  it. 
Pieces  of  hard  wood,  of  loaf-sugar,  and  brittle  mineral  substances, 
are  split  in  two,  or  shivered  to  pieces,  by  an  intense  charge  of  a 
battery.  But  good  conductors  of  much  breadth  are  not  thus 


UNIVERSITY 


ROUTES    OF    DISCHARGE.  419 

affected.  The  charge,  as  it  is  transmitted,  passes  over  the  whole 
body,  instead  of  being  concentrated  in  any  one  line.  But  if 
liquids  which  are  good  conductors  are  closely  confined  on  every 
side,  they  show  that  a  violent  expansion  is  produced  by  a  dis- 
charge. Thus,  when  a  charge  is  sent  through  water  confined  in 
a  small  glass  tube  or  ball,  the  glass  is  shattered  to  pieces ;  and 
mercury  in  a  thick  capillary  tube  is  expanded  with  a  force  suffi- 
cient to  splinter  the  glass. 

668.  Different  Routes  of  Discharge. — If  two  or  more 
circuits  are  opened  at  once  between  the  two  coatings  of  a  charged 
jar  or  battery,  the  discharge  will  take  one  or  another,  or  divide 
between  them,  according  to  circumstances.     If  the  circuits  are 
alike  except  in  length,  the  discharge  will  follow  the  shorter.     If 
they  differ  only  in  conducting  quality,  the  electricities  will  take 
the  best  conductor.     If  the  circuits  are  interrupted,  and  in  all 
respects  alike,  except  that  the  conductors  of  one  are  pointed  at  the 
interruptions,  and  of  the  others  not  pointed,  the  discharge  will 
follow  the  line  which  has  pointed  conductors.    If  the  circuits  are 
very  attenuated  (as  very  fine  wire,  or  threads  of  gold-leaf),  the 
charge  is  liable  to  divide  among  them. 

669.  Chemical  Effects. — These  are  various:  combustion  of 
inflammable  bodies ;  oxydation,  fusion,  and  combustion  of  metals  ; 
separation  of  compounds  into  their  elements  ;  reunion  of  elements 
into  compounds. 

Ether  and  alcohol  may  be  inflamed  by  passing  the  electric 
spark  through  them  ;  phosphorus,  resin,  and  other  solid  combus- 
tible bodies,  may  be  set  on  fire  by  the  same  means;  gunpowder 
and  the  fulminating  powders  may  be  exploded,  and  a  candle  may 
be  lighted.  Gold-leaf  and  fine  iron  wire  may  be  burned,  by  a 
charge  from  the  battery.  Wires  of  lead,  tin,  zinc,  copper,  plati- 
num, silver,  and  gold,  when  subjected  to  the  charge  of  a  very 
large  battery,  are  burned,  and  converted  into  oxides. 

The  same  agent  is  also  capable  of  restoring  these  oxides  to 
their  simple  forms.  Water  is  decomposed  into  its  gaseous  ele- 
ments, and  these  elements  may  again  be  reunited  to  form  water. 
By  passing  a  great  number  of  electric  charges  through  a  confined 
portion  of  air,  the  oxygen  and  nitrogen  are  converted  into  nitric 
acid.  The  ozone  which  is  almost  always  perceived  in  connection 
with  electrical  experiments  is  to  be  considered  as  one  of  the 
chemical  effects  of  electricity. 

Galvanic  electricity  is  a  form  of  this  agent  much  better  adapted 
than  frictional  electricity  to  produce  chemical  as  well  as  magnetic 
effects. 

670.  Magnetic  Effects. — It  is  difficult  to  cause  deflections 


420      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

of  a  magnetic  needle  on  account  of  the  high  potential  of  f  rictional 
electricity,  which  renders  it  necessary  to  take  special  pains  to 
render  the  insulation  of  the  helix  of  the  galvanometer  very  per- 
fect. A  galvanometer  of  400  or  500  turns  of  fine  wire  very  per- 
fectly insulated  by  layers  of  insulating  substance  of  appreciable 
thickness,  will  give  deflections  of  the  needle  when  connected  with 
the  two  discharging  knobs  of  a  Holtz  machine. 

A  steel  needle  may  be  magnetized  by  passing  the  electricity 
through  a  properly  insulated  helix. 

The  galvanometer  and  its  use  will  be  explained  in  Part  IX. 

671.  Physiological    Effects. — The   shock  experienced  by 
the  animal  system,  when  the  charge  of  a  jar  passes  through  it,  has 
been  already  mentioned. 

A  slight  charge  of  the  Leyden  jar,  passed  through  the  body 
from  one  hand  to  the  other,  affects  only  the  fingers  or  the  wrists  ; 
a  stronger  charge  convulses  the  large  muscles  of  the  arms  ;  a 
still  greater  charge  is  felt  in  the  breast,  and  becomes  somewhat  pain- 
ful. The  charge  of  a  large  battery  is  sufficient  to  destroy  life,  if 
it  be  sent  through  the  vital  organs.  By  connecting  the  chains 
which,  are  attached  to  the  jar  with  insulating  handles,  it  is  easy  to 
pass  shocks  through  any  particular  joint,  muscle,  or  other  part  of 
the  body,  as  is  frequently  done  for  medical  purposes. 

The  charge  may  be  passed  through  a  great  number  of  persons 
at  the  same  time,  in  which  case  those  at  the  centre  of  the  chain 
will  receive  a  less  severe  shock  than  those  near  the  ends.  Hun- 
dreds of  individuals,  by  joining  hands,  have  received  the  shock  at 
once,  though  there  is  more  difficulty  in  passing  a  given  charge  as 
the  number  is  increased. 

If  the  spark  is  taken  by  a  person  from  the  prime  conductor, 
the  quantity  is  not  sufficient,  unless  the  conductor  is  of  extraordi- 
nary size,  to  produce  what  is  called  the  shock  ;  a  pricking  sensa- 
tion in  the  flesh  where  the  spark  strikes,  and  a  slight  spasm  of  the 
muscle,  is  all  that  is  noticeable.  A  person  may  make  his  own 
body  a  part  of  the  prime  conductor  by  standing  on  an  insulating 
stool— that  is,  a  stool  having  glass  legs,  and  touching  the  conduc- 
tor of  the  machine.  This  occasions  no  sensation  at  all,  except 
what  arises  from  the  movement  of  the  hair,  in  yielding  to  the  re- 
pellency  of  the  fluid.  If  another  person  takes  the  spark  from  him, 
the  prick  is  more  pungent,  as  the  quantity  is  larger  than  in  the 
prime  conductor  alone. 

672.  Velocity  of  Electricity. — This  is  so  great  that  no 
appreciable  time  is  occupied  in  any  case  of  discharge.     It  has  been 
determined  that  the  duration  of  the  lightning  flash  is  less  than 

second.     Wheatstone  found  the  duration  of  the  spark 


ELECTRICITY    IN    THE  AIR.  421 


of  a  Leyden  jar  to  be  ^jioij-  °^  a  second.  The  duration  of  the 
spark  is  greater  as  the  resistance  of  the  medium  is  greater,  as  the 
striking  distance  is  greater,  and  as  the  charge  is  increased. 

Wheatstone  devised  an  ingenious  method  of  measuring  the 
time  in  which  electricity  passes  over  a  wire  only  half  a  mile  long. 
The  wire  was  so  arranged  that  three  interruptions,  one  near  each 
end,  and  one  in  the  middle  of  the  wire,  were  brought  side  by  side. 
When  the  discharge  of  a  jar  was  transmitted,  the  sparks  at  these 
interruptions  were  seen  by  reflection  in  a  swiftly  revolving  mirror. 
An  exceedingly  small  difference  of  time  between  the  passage  of 
those  interruptions  could  be  easily  perceived  by  the  displacement 
of  the  sparks  as  seen  in  the  whirling  mirror.  The  amount 
of  observed  displacement  and  the  known  rate  of  revolution  of 
the  mirror,  would  furnish  the  interval  of  time  occupied  by  the 
electricity  in  passing  from  one  interruption  to  the  next.  By 
a  series  of  experiments,  Wheatstone  arrived  at  the  conclusion 
that,  on  copper  wire,  one-fifteenth  of  an  inch  in  diameter,  electri- 
city moves  at  the  rate  of  288,000  miles  per  second,  a  velocity  much 
greater  than  that  of  light. 

Galvanic  electricity  moves  very  much  slower.  Its  rate  on  iron 
wire,  of  the  size  usually  employed  for  telegraph  lines,  is  about 
16,000  miles  per  second. 

No  definite  velocity  can  be  assigned  to  electricity,  except  under 
specified  conditions.  As  we  cannot  say  what  will  be  the  velocity 
of  sound,  or  of  light,  without  specifying  the  medium  traversed,  so 
in  the  case  of  electrical  currents  we  must  consider  the  material  of 
the  conductor,  its  cross-section,  its  length,  and  the  surrounding 
inductive  influences. 


CHAPTER    V. 

ATMOSPHERIC    ELECTRICITY— THUNDER-STORMS. 

673.  Electricity  in  the  Air. — The  atmosphere  is  always 
more  or  less  electrified,  sometimes  positively,  sometimes  negatively. 
This  fact  is  ascertained  by  several  different  forms  of  apparatus. 
For  the  lower  strata,  it  is  sufficient  to  elevate  a  metallie  rod  a  few 
feet  in  length,  pointed  at  the  top,  and  insulated  at  the  bottom. 
With  the  lower  extremity  is  connected  an  electroscope,  which  in- 
dicates the  presence  and  intensity  of  the  electricity.  For  experi- 
ments on  the  electricity  of  higher  portions,  a  kite  is  employed, 


422      PRICTIONAL  OR  STATICAL  ELECTRICITY. 

with  the  string  of  which  is  intertwined  a  fine  metallic  wire.  The 
lower  end  of  the  string  is  insulated  by  fastening  it  to  a  support  of 
glass,  or  by  a  cord  of  silk.  If  a  cloud  is  near  the  kite,  the  quantity 
of  electricity  conveyed  by  the  string  may  be  greatly  increased,  and 
even  become  dangerous.  Cavallo  received  a  large  number  of  severe 
shocks  in  handling  the  kite-string;  and  Richmann,  of  St.  Peters- 
burgh,  was  killed  by  a  discharge  of  electricity  which  came  down 
the  rod  which  he  had  arranged  for  his  experiments,  but  which  was 
not  provided  with  a  conductor  near  by  it,  for  taking  off  extra 
charges. 

The  electricity  of  the  atmosphere  is  most  developed  when  hot 
dry  weather  succeeds  a  series  of  rainy  days,  or  the  reverse ;  and 
during  a  single  day,  the  air  is  most  electrical  when  dew  is  begin- 
ning to  form  before  sunset,  or  when  it  begins  to  exhale  after  sun- 
rise. In  clear,  steady  weather,  the  electricity  is  generally  positive ; 
but  in  falling  or  stormy  weather,  it  is  frequently  changing  from 
positive  to  negative,  and  from  negative  to  positive. 

674.  Thunder-Storms. — Thunder-clouds  are,  of  all  atmos- 
pheric bodies,  the  most  highly  charged  with  electricity ;  but  all 
single,  detached,  or  insulated  clouds  are  electrified  in  greater  or  less 
degrees,  sometimes  positively  and  sometimes  negatively.  When, 
however,  the  sky  is  completely  overcast  with  a  uniform  stratum  of 
clouds,  the  electricity  is  much  feebler  than  in  the  single  detached 
masses  before  mentioned.  And,  since  fogs  are  only  clouds  near 
the  surface  of  the  earth,  they  are  subject  to  the  same  conditions  : 
a  driving  fog,  of  limited  extent,  is  often  highly  electrified. 

Thunder-storms  occur  chiefly  in  the  hottest  season  of  the  year, 
and  after  midday,  and  are  more  frequent  and  violent  in  warm  than 
in  cold  countries.  They  never  occur  beyond  75°  of  latitude — sel- 
dom beyond  65°.  In  the  New  England  States  they  usually  come 
from  the  west,  or  some  westerly  quarter. 

The  storm  itself,  including  everything  except  the  electrical 
appearances,  is  supposed  to  be  produced  in  the  same  manner  as 
other  storms  of  wind  and  rain ;  and  the  electricity  is  developed 
by  the  rapid  condensation  of  watery  vapor,  and  by  friction.  Elec- 
tricity is  not  to  be  regarded  as  the  cause,  but  as  a  consequence  or 
concomitant  of  the  storm.  But  the  precipitation  of  vapor  must  be 
sudden  and  copious,  since  when  the  process  is  slow,  too  much  of 
the  electricity  evolved  would  escape  to  allow  of  the  requisite  accu- 
mulation. Also,  if  a  storm-cloud  is  of  great  extent,  it  is  not  likely 
to  be  highly  electrified,  because  the  opposite  electricities,  which 
may  be  developed  in  different  parts  of  it,  have  opportunity  to 
mingle  and  neutralize  ;  and  points  of  communication  with  the 
earth  will  here  and  there  occur.  Clouds  of  rapid  formation, 


LIGHTNING.  423 

violent  motion,  and  limited  extent,  are  therefore  most  likely  to  be 
thunder-clouds. 

675.  Lightning. — When  a  cloud  is  highly  charged,  it  oper- 
ates inductively  on  other  bodies  near  it,  such  as  other  clouds,  or 
the  earth.     Hence,  discharges  will  occur  between  them.    Light- 
ning passes  frequently  between  two  clouds,  or  even  between  two 
parts  of  the  same  cloud,  in  which  opposite  electricities  are  so 
rapidly  developed  that  they  cannot  mingle  by  conduction.     But, 
in  general,  the  discharges  of  lightning  take  place  between  the 
electrified  cloud  and  the  earth,  whose  nearer  part  is  thrown  into 
the  opposite  electrical  state  by  induction.    It  is  supposed  that,  in 
some  instances,  a  discharge  occurs  between  two  distant  clouds  by 
means  of  the  earth,  which  constitutes  an  interrupted  circuit  be- 
tween them.     The  crinkled  form  of  the  path  of  lightning  is 
explained  in  the  same  way  as  that  of  the  spark  from  the  machine, 
and   the  thunder  is  caused   by  the  simultaneous  rupture  and 
collapse  of  air  in  all  parts  of  the  line  of  discharge.     The  words 
chain-lightning,  sheet-lightning,  and  heat-lightning,  are  supposed 
not  to  indicate  any  real  differences  in  the  lightning  itself,  but 
only  in  the  circumstances  of  the  person  who  observes  it.     If  the 
crinkled  line  of  discharge  is  seen,  it  is  chain  or  fork  lightning;  if 
only  the  light  which  proceeds  from  it  is  noticed,  it  is  sheet-light- 
ning  ;  if,  in  the  evening,  the  thunder-storm  is  so  far  distant  that 
the  cloud  cannot  be  seen,  nor  the  thunder  heard,  but  only  the 
light  of  its  discharges  can  be  discerned  in  the  horizon,  it  is  fre- 
quently called  heat-lightning. 

676.  Identity  of  Lightning  and  Electrical  Discharges. 

— Franklin  was  the  first  to  point  out  the  resemblances  between 
the  phenomena  of  lightning  and  those  of  frictional  electricity.  He 
was  also  the  first  to  propose  the  performance  of  electrical  experi- 
ments by  means  of  electricity  drawn  from  the  clouds.  The  points 
of  resemblance  named  by  Franklin  were  these :  1.  The  crinkled 
form  of  the  path.  2.  Both  take  the  most  prominent  points.  3. 
Both  follow  the  same  materials  as  conductors.  4.  Both  inflame 
combustible  substances.  5.  They  melt  metals  in  attenuated  forms. 
6.  They  fracture  brittle  bodies.  7.  Both  have  produced  blind- 
ness. 8.  Both  destroy  animal  life.  9.  Both  affect  the  magnetic 
needle  in  the  same  manner.  In  1752,  he  obtained  electricity  from 
a  thunder-cloud  by  a  kite,  and  charged  jars  with  it,  and  performed 
the  usual  electrical  experiments. 

677.  Lightning-Rods. — Franklin  had  no  sooner  satisfied 
himself  of  the  identity  of  electricity  and  lightning  than,  with  his 
usual  sagacity,  he  conceived  the  idea  of  applying  the  knowledge 


424      PRICTIONAL    OR    STATICAL    ELECTRICITY. 

acquired  of  the  properties  of  the  eleotric  fluid  so  as  to  provide 
against  the  dangers  of  thunder-storms.  The  conducting  power  of 
metals,  and  the  influence  of  pointed  bodies  to  transmit  the  fluid, 
naturally  suggested  the  structure  of  the  lightning-rod.  The 
experiment  was  tried,  and  has  proved  completely  successful;  and 
probably  no  single  application  of  scientific  knowledge  ever  secured 
more  celebrity  to  its  author. 

678.  Rules  for  the  Protection  of  Buildings.— The  fol- 
lowing rules  are  derived  from  the  "  Report  of  the  Lightning  Rod 
Conference,"  consisting  of  delegates  from  the  following  British 
societies:  Meteorological  Society,  the  Royal  Institute  of  British 
Architects,  the  Society  of  Telegraph  Engineers  and  Electricians, 
and  the  Physical  Society  : 

Points. — The  point  of  the  upper  terminal  should  be  a  cone 
whose  height  is  equal  to  the  radius  of  its  base.  A  foot  below  this 
cone  a  copper  ring  should  be  screwed  and  soldered  to  the  rod,  in 
which  ring  should  be  soldered  three  or  four  sharp  copper  points, 
about  six  inches  long,  gilded  or  nickel-plated. 

Number  of  Upper  Terminals. — There  can  be  no  rule  as  to  the 
proper  number  of  terminals.  The  space  protected  is  assumed  to 
be  a  conical  space,  the  radius  of  whose  base  is  equal  to  the  height 
of  the  terminal.  All  portions  of  a  building  within  such  cone  will 
be  reasonably  protected.  Chimneys  should  always  be  protected 
by  terminals  on  account  of  the  conducting  power  of  the  ascending 
heated  gases  and  vapors. 

Attachment. — The  rod  should  never  be  insulated  from  the 
building.  It  should  be  supported  by  metal  straps,  or  fastenings, 
of  the  same  material  as  the  rod,  of  such  form  as  not  to  compress 
or  distort  the  rod,  and  in  such  manner  as  to  allow  for  expansion 
and  contraction  of  the  rod  from  changes  of  temperature. 

Ornamental  Metal  Work. — All  vanes,  finials,  ridge-ironwork, 
metal  roofing,  and  other  metal  masses  upon  a  building  should  be 
connected  with  the  conductor. 

Material  for  Rod. — Copper  is  recommended  as  eventually 
cheaper  than  iron,  as  being  lighter,  and  as  being  more  durable. 
It  is  more  likely  to  be  stolen,  in  places  where  buildings  are  not 
occupied  during  the  whole  year. 

Cross-section  of  Conductor. — The  minimum  dimensions  author- 
ized by  the  Report  are  : 

Material.  Form.  Section.  Weight 

Sq.  in.  per  ft. 

Copper Rope,  .  10  6  oz. 

"      Rod,                      .11  7oz. 

" Tape  (f  x£),  .09  6  oz. 

Iron Rod,  .64  35  oz. 


PROTECTION    BY    LIGHTNING-RODS.  425 

The  advantages  of  rods  are  their  durability,  and  rigidity  when 
used  for  terminals.  The  disadvantages  are  the  numerous  joints 
required,  and  the  difficulty  of  bending  to  conform  to  the  outlines 
of  the  building. 

Tapes  are  good  since  they  are  flexible,  and  the  joints  may  be 
made  very  perfect  by  riveting  and  then  soldering.  No  sharp 
bends  should  be  allowed.  The  copper  used,  in  any  form,  must 
have  a  conductivity  of  not  less  then  90  per  cent,  that  of  pure 
copper. 

Hopes  should  be  made  of  wires  of  not  less  than  No.  12  B.  W.  G., 
about  TVo  of  an  inch  diameter.  Iron  should  only  be  used  in  the 
form  of  rod,  either  square  or  round. 

Joints. — Every  joint  must  have  bright  metallic  surfaces  in  con- 
tact, and  after  being  screwed,  or  riveted,  should  be  entirely 
covered  with  solder. 

Curves. — The  rod  should  not  be  bent  abruptly.  In  no  case 
•should  the  length,  measured  along  a  curve,  be  more  than  half  as 
long  again  as  the  chord  subtending  such  curve. 

Metal  Pipes. — The  conductor  should  be  connected  with  all 
pipes  and  other  large  masses  of  metal  within  the  building,  except 
gas  pipes.  An  electrically  defective  joint  in  a  gas  pipe  might 
result  in  an  explosion. 

Earth  Connection. — The  lower  end  of  the  conductor  must  be 
placed  in  permanent  moisture.  The  earth  connection  should  not 
be  poorer  than  that  afforded  by  a  copper  plate  3  feet  square  and 
-^  inch  thick,  buried  in  permanently  wet  earth,  and  surrounded 
by  coke  or  charcoal.  If  iron  is  used  for  the  rod,  a  galvanized  iron 
plate  of  similar  dimensions  should  be  used  in  like  manner. 

Examination. — The  protective  system  of  rods  should  be  regu- 
larly inspected  and  tested  for  conductivity,  as  breaks  may  occur, 
either  above  ground  or  below,  which  would  not  only  render  the 
rod  useless  as  a  protection,  but  positively  a  source  of  additional 
danger. 

Painting. — The  rods  should  be  protected  by  paint,  except 
where  the  terminals  require  a  bright  metallic  surface  to  be  ex- 
posed. 

679.  In  what  way  Lightning-Rods  Afford  Protection. 

— Lightning-rods  are  of  service,  not  so  much  in  receiving  a  dis- 
charge when  it  comes,  as  in  diminishing  the  number  of  discharges 
in  their  vicinity.  The  sharp  points  upon  the  copper  ring  con- 
tinually carry  on  a  silent  communication  between  the  two  electri- 
cities, which  are  attracting  each  other,  one  in  the  cloud,  the  other 
in  the  earth ;  so  that  a  village  well  furnished  with  rods  has  few 
discharges  of  lightning  in  it.  All  tall  pointed  objects,  like  spires 


426      FRICTIONAL  OR  STATICAL  ELECTRICITY. 

of  churches  and  masts  of  ships,  exert  a  similar  influence,  though 
in  a  less  degree,  because  not  so  good  conductors. 

During  a  thunder-storm,  or  immediately  after  it,  if  a  person 
can  be  near  the  top  of  a  high  rod,  he  will  sometimes  hear  the  hiss- 
ing sound  of  electricity  escaping  from  it,  as  from  a  point  attached 
to  the  prime  conductor  of  a  machine.  In  the  same  circumstances, 
if  it  were  quite  dark,  he  would  probably  see  the  brushes  or  stars 
of  light  on  the  points.  The  statement  of  Caesar  in  his  Commen- 
taries, "that  the  points  of  the  soldiers'  darts  shone  with  light  in 
the  night  of  a  severe  storm,"  probably  refers  to  the  visible  escape 
of  electricity  from  the  weapons  as  from  lightning-rods. 

680.  Protection  of  the  Person. — Silk  dresses  are  some- 
times worn  with  the  view  of  protection,  by  means  of  the  insu- 
lation they  afford.     They  cannot,  however,  be  deemed  effectual 
unless  they  completely  envelop  the  person ;  for  if  the  head  and  the 
extremities  of  the  limbs  are  exposed,  they  will  furnish  so  many 
avenues  as  to  render  the  insulation  of  the  other  parts  of  the 
system  of  little  avail.     The  same  remark  applies  to  the  supposed 
security  that  is  obtained  by  sleeping  on  a  feather  bed.    Were  the 
person  situated  within  the  bed,  so  as  to  be  entirely  enveloped  by 
the  feathers,  they  would  afford  some  protection  ;  but  if  the  person 
be  extended  on  the  surface  of  the  bed,  in  the  usual  posture,  with 
the  head  and  feet  nearly  in  contact  with  the  bedstead,  he  would 
rather  lose  than  gain  by  the  non-conducting  properties  of  the  bed, 
since,  being  a  better  conductor  than  the  bed,  the  charge  would 
pass  through  him  in  preference  to  that.     If  the  bedstead  were  of 
iron,  its  conducting  quality  would  probably  be  a  better  protection 
than  the  insulating  property  of  the  feathers,  since,  by  taking  the 
charge  itself,  it  would  keep  it  away  from  the  person.    So,  a  man's 
garments  soaked  with  rain  have  been  known  to  save  his  life,  being 
a  better  conductor  than  his  body.     Animals  under  trees  are  pecu- 
liarly exposed,  because  the  trees  by  their  prominence  are  liable  to 
be  the  channels  of  communication  for  the  electric  discharge,  and 
the  animal  body,  so  far  as  it  reaches,  is  a  better  conductor  than 
the  tree.     Tall  trees,  however,  situated  near  a  dwelling-house, 
furnish  a  partial  protection  to  the  building,  being  both  better 
conductors  than  the  materials  of  the  house,  and  having  the  ad- 
vantage of  greater  elevation. 

681.  How  Lightning  Causes  Damage. — The  word  strike, 
which  is  used  with  reference  to  lightning,  conveys  no  correct  idea 
of  the  nature  of  the  movement  of  electricity,  or  of  the  injury  which 
it  causes.     One  kind  of  electricity,  developed  in  a  cloud,  causes 
the  other  to  be  accumulated  by  induction  in  the  part  of  the  earth 
nearest  to  it.      These  electricities  strongly  attract  each  other ; 


HOW     LIGHTNING    CAUSES    DAMAGE.          427 

consequently,  that  in  the  earth  presses  upward  into  all  prominent 
conducting  bodies  toward  the  other;  and,  if  those  bodies  are 
numerous,  high,  the  best  of  conductors,  and  terminated  by  points, 
the  electricity  will  flow  off  from  them  abundantly,  and  mingle  with 
its  opposite  in  the  air  above  ;  and  thus  discharges  are  in  a  great 
degree  prevented.  But  if  these  channels  for  silent  communication 
are  not  furnished,  the  quantity  of  electricity  will  increase,  till  the 
strength  of  attraction  becomes  so  great  that  the  fluid  will  break 
its  way  through  the  air,  usually  from  some  prominent  object,  as  a 
building  or  tree,  and  thus  the  union  of  the  two  electricities  takes 
place.  The  building  or  tree  in  this  case  is  said  to  be  struck  ly 
lightning  ;  it  is  rent,  or  otherwise  injured,  by  the  great  quantity 
of  electricity  which  passes  violently  through  it,  in  an  inconceivably 
short  space  of  time.  The  effects  produced  are  exactly  like  those 
caused  by  discharges  of  the  electrical  battery,  on  a  greatly  enlarged 
scale.  The  charge  of  a  large  battery,  taken  through  the  body  in 
the  usual  way,  would  prostrate  a  person  by  the  violence  of  the 
shock ;  but  the  same  charge,  if  allowed  to  occupy  a  few  seconds 
in  passing  by  means  of  a  point,  would  not  be  felt  at  all. 

Fulgurites  are  tubes  of  silicious  matter  formed  in  the  ground, 
where  lightning  has  struck  in  sandy  soil,  and  melted  the  sand 
around  its  path  towards  the  conducting  moist  earth  below. 


PART    IX. 

ELECTRICITY, 


CHAPTER    I. 

THE  GALVANIC  CURRENT,  AND  APPARATUS  FOR  PRODUCING  IT. 

682.  Electricity  Developed  by  Chemical  Action. — In  a 

glass  vessel  (Fig.  369)  containing  a  mixture  of  one  part  of  sul- 
phuric acid  and  seven  or  eight  parts  of 
water,  put  two  plates,  one  of  copper,  C, 
and  the  other  of  zinc,  Z,  to  each  of  which 
is  soldered  a  copper  wire.  On  bringing 
the  extreme  ends  of  the  wires  together,  a 
feeble  flow  of  electricity  will  take  place 
through  the  wires,  the  plates,  and  the 
liquid,  as  may  be  shown  by  the  peculiar 
taste  or  sensation  which  is  perceived  on 
placing  one  terminal  wire  upon  and  the 
other  beneath  the  tongue,  or  better  still, 
by  deflections  of  the  magnetic  needle  arranged  as  described  here- 
after. Electricity  thus  produced  is  called  galvanic  or  voltaic, 
from  Galvani  and  Volta,  two  Italian  philosophers,  who  made  the 
first  discoveries  of  importance  in  this  branch  of  science.  It  is  also 
called  dynamical  electricity  because  of  its  constant  flow  through 
the  conductor. 

In  using  the  terms  flow  and  current  the  student  must  keep  in 
mind  that  there  is  no  transfer  of  any  substance,  no  flow  nor  cur- 
rent in  the  sense  in  which  we  use  the  terms  when  applied  to 
water  ;  such  words  are  convenient  to  express  the  transmission  of 
the  molecular  disturbances,  vibrations,  or  tensions,  due  to  electric 
action,  just  as  we  may  speak  of  the  flow  of  heat  through  a  dia- 
thermal  substance. 

683.  Definitions. — An  element  or  cell  is  a  jar  containing  any 
arrangement  of  substances  for  the  purpose  of  producing  a  current 


VOLTA'S     PILE. 


FIG.  370. 


of  electricity.  ,A  battery  is  a  number  of  elements  properly  con- 
nected with  each  other. 

The  poles  or  electrodes  of  a  cell  or  battery  are  the  extremities 
of  the  wires  where  the  electricities  appear. 

The  circuit  is  the  path  or  conductor  provided  for  the  flow  of 
the  current — that  is,  the  liquid,  the  plates,  and  the  wires.  The 
circuit  is  said  to  be  closed  when  the  wires  are  joined,  so  that  there 
is  a  flow  of  the  current ;  when  they  are  separated,  the  current 
ceases,  and  the  circuit  is  said  to  be  broken,  or  to  be  open. 

When  the  circuit  is  broken  it  will  be  found  that  the  outer  end 
of  the  wire  attached  to  the  copper  plate  will  be  of  higher  poten- 
tial than  the  outer  end  of  the  wire  attached  to  the  zinc.  The 
current  is  said  to- flow  from  the  copper  plate,  through  the  wire 
conductor  to  the  zinc  plate,  and  from  the  zinc  plate  through  the 
the  liquid  to  the  copper  plate. 

684.  Volta's  Pile.— In  this  form  of  current  generator,  the 
origin  of  the  ordinary  battery  of  cells,  a  large  number  of  alternate 
disks  of  copper,  zinc,  and  cloth  moistened  with  very  dilute  acid, 
are  used,  as  shown  in  Fig.  370.     If  this  pile  be  set  upon  an  insu- 
lating plate  it  will  be  found,  upon  testing  with 

the  proof-plane  and  electroscope,  that  the 
opposite  ends  are  charged  with  opposite  elec- 
tricities, the  potential  increasing  from  the 
middle,  or  neutral  point,  towards  each  end. 
If  the  terminal  disk  of  copper  be  connected 
with  the  terminal  zinc  by  a  conducting  wire,  a 
current  will  flow  as  in  the  battery  of  cells,  the 
moistened  cloth  disks  corresponding  in  this 
case  to  the  liquid  in  the  ordinary  cell. 

685.  Electricity  Due  to  Contact. — If 

a  strip  of  copper  be  soldered  to  a  strip  of  zinc  the  copper  will  be- 
come negatively  and  the  zinc  positively  charged,  as  may  be  shown 
by  touching  the  lower  plate  of  a  condensing  electroscope  with  the 
copper  end  of  the  couple,  the  zinc  being  held  in  the 
hand.  If  a  rectangle  be  formed  of  copper  and  zinc, 
or  other  dissimilar  metals,  as  in  Fig.  371,  and  a 
needle,  one  half  c  made  of  gilt  paper  and  the  other 
.1  ~  II.  nalf  d  of  lac>  be  Pr°Perly  suspended  within,  then 
upon  charging  c  with  positive  electricity,  it  will 
turn  towards  the  copper  side  of  the  junction  b, 
while  a  negative  charge  will  turn  it  towards  the 
zinc  side  of  b.  From  these,  and  similar  experi- 
ments, it  is  concluded  that  electric  decomposition  is  the  result 
of  contact  of  dissimilar  substances. 


FIQ.  371. 
a, 


430 


DYNAMICAL    ELECTRICITY. 


FIG.  372. 


Z 


686.  The  Cell  of  Two  Fluids.— An  element  of  copper, 
zinc,  and  dilute  acid,  already  described,  soon  loses  its  efficiency, 
because  of  the  deposit  of  hydrogen  upon  the  copper  plate,  the 
result  of  the  chemical  decomposition  due  to  electric  action,  which 
deposit  not  only  acts  as  a  film  of  low  conductivity,  but  also  tends 
to  cause  a  reverse  current  through  the  circuit. 

In  improved  batteries,  by  which  a  constant  flow  of  electricity 
may  be  maintained  for  a  considerable  length  of  time,  two  liquids 
•  are  employed,  and  generally  some  other  substance  than  copper  for 
one  of  the  metals.  The  liquids  must  be  separated  by  some  porous 
substance,  which  shall  prevent  them  from  mingling,  and  at  the 
same  time,  being  saturated  by  the  liquid,  shall  not  interrupt  the 
necessary  moist  communication  between  the  metals. 

687.  Constant  Batteries. — Batteries  composed  of  cells  con- 
taining two  liquids  are  called  constant,  because  their  action  con- 
tinues for  so  long  a  time  without  sensible  abatement. 

Daniell's  Cell. — One  of  the  most  constant  of  all  the  forms  of 
cell  yet  devised  is  that  of  Daniell.  This  con- 
sists of  a  copper  cup  c  (Fig.  372),  within  which 
is  a  smaller  cylinder  of  porous  earthenware  p 
containing  a  rod  of  zinc  z.  The  outer  copper 
vessel  is  filled  with  a  saturated  solution  of 
cupric  sulphate,  and  the  porous  cup  is  charged 
with  dilute  sulphuric  acid.  As  the  strength 
of  the  cupric  sulphate  decreases  gradually, 
when  the  current  flows,  crystals  must  be  added 
from  time  to  time. 

Groves  Battery. — One  element  is  shown  in 

Fig.  373,  which  represents  a  glass  jar  containing  a  hollow  cylinder 

of  zinc,  which  has  a  narrow  opening  on  one  side  from  top  to  bot- 
tom, that  the  liquid  in  which  it  is  placed  may 

circulate  freely  within  it.     Within  the  zinc 

is  a  cylindrical  cup  of  porous  earthenware, 

and  within  that  is  suspended  a  lamina  of 

platinum.     One  of  the   circuit  wires  is  in 

metallic  communication  with  the  zinc,  and 

the  other  with  the  platinum,  by  means  of  the 

binding  screws  at  the  top.     The  earthen  cup 

is  now  filled  with  strong  nitric  acid,   while 

the  space  outside  of  it,  in  which  the  zinc  is 

placed,  contains  dilute  sulphuric  acid. 

Bunsen's  Battery  is  the  same  as  Grove's,  except  that  in  it  a 

cylinder  of  carbon  is  used  instead  of  a  leaf  of  platinum,  on  account 


FIG.  37B. 


AMALGAMATION     OF    ZINC.  431 

of  the  expense  of  the  latter.     It  is  very  generally  employed  in 
telegraphy.     Fig.  374  is  a  Bunsen  battery  of  ten  cells. 


FIG.  374. 


Gravity  Batteries. — As  the  porous  cups  are  gradually  cracked 
and  impaired  by  crystallization  within  their  substance,  or  by  a 
deposit  of  copper  in  cupric  sulphate  cells,  the  difference  in 
density  of  different  solutions  has  been  relied  upon  to  maintain 
separation  of  the  liquids.  In  one  form  of  gravity  cell  the  copper 
plate  is  a  horizontal  disk  at  the  bottom  of  a  jar  ;  upon  this  is  a 
saturated  solution  of  cupric  sulphate,  above  which  is  a  solution 
of  zinc  sulphate,  and  immersed  in  this,  at  the  top  of  the  jar,  is  a 
horizontal  disk  or  grating  of  zinc.  Such  batteries  gradually  be- 
come impaired  by  diffusion  of  the  two  liquids,  unless  special 
devices  be  employed  to  prevent  such  result. 

It  would  extend  the  subject  beyond  our  limits  to  attempt  a 
description  of  the  numberless  forms  of  cells  in  use,  or  which  have 
been  proposed;  it  is  also  unnecessary  here  to  enter  upon  a  dis- 
xmssion  of  the  chemical  decompositions  and  exchanges  \vhich 
result  when  a  current  flows. 

688c  Amalgamation  of  Zinc. — Pure  zinc  is  not  dissolved 
by  dilute  sulphuric  acid,  while  commercial  zinc  is  rapidly  acted 
upon. 

When  a  commercial  zinc  plate  is  used  in  a  cell  the  impurities 
present,  either  lead  or  iron,  give  rise  to  local  currents  from  one  part 
of  the  plate  to  another,  and  not  only  cause  a  rapid  solution  of  the 
plate,  but  by  such  currents  impair  the  efficiency  of  the  cell. 

If  the  zinc  plate  be  well  amalgamated,  by  first  cleaning  with 
dilute  sulphuric  or  hydrochloric  acid,  and  then  applying  a  little 
mercury  to  the  bright  surfaces,  a  film  of  zinc  dissolved  in  mercury 
spreads  over  the  whole  plate,  acting  as  a  surface  of  pure  zinc  when 
in  contact  with  the  liquid  of  the  cell. 


432  DYNAMICAL     ELECTRICITY. 

689.  Electromotive  Force. — The  force  which  causes  a  cur- 
rent to  traverse  a  conducting  medium,  such  as  the  conducting  wire 
of  the  battery  circuit  or  the  liquids  within  the  cells,  is  termed  the 
electromotive  force  ;  it  is  this  force  which  causes  the  differences  of 
potential  upon  contact  of  heterogeneous  substances. 

Electromotive  force  is  independent  of  the  amount  of  surface 
of  contact.  A  Daniell  cell  with  plates  of  one  square  inch  each 
has  the  same  electromotive  force  as  a  similar  cell  having  plates 
1000  square  inches  each  ;  for  the  electromotive  force  is  measured 
by  the  differences  of  potential,  and  the  difference  of  potential 
being  the  same  at  every  elementary  unit  of  surface  of  contact  of 
like  substances,  the  electromotive  force  must  remain  constant, 
whatever  change  may  be  made  in  the  areas  of  these  surfaces. 

The  practical  unit  of  electromotive  force  is  the  volt,  and  is  very 
nearly  equal  to  that  of  a  Daniell  cell,  which  is  usually  referred  to- 
ns the  standard. 

The  following  table  gives  the  electromotive  force  of  the  dif- 
ferent cells  mentioned,  amalgamated  zinc  being  used  in  each  : 


Daniell. 

Sul.  Acid,  Jy, 

Saturated  Cupric  Sulphate. 

Copper. 

1.079  volts. 

<« 

<•                   «                    X 

Saturated  Cupric  Nitrate. 

*« 

1.000     f< 

Grove  .  . 

Sul.  Acid,  T\, 

Nitric  Acid  (Fuming). 

Platinum. 

1.956     " 

Bunsen  . 

"      "        " 

f<                  («                          X 

Carbon. 

1.734     " 

690.  Resistance. — The  consideration  of  electromotive  force 
directs  attention  at  once  to  the  resistances  which  that  force  over- 
comes in  urging  the  current  through  a  medium.  The  laws 
governing  the  resistance  of  a  conductor  are  : 

1.  The  resistance  of  a  conductor  depends  upon  the  material  of 
which  it  is  made. 

Among  metals  silver  and  copper  offer  the  least  resistance, 
while  the  resistance  of  liquids  is  very  great,  being  many  million 
times  that  of  pure  silver. 

2.  The  resistance  varies  inversely  as  the  cross-section  of  the  con- 
ductor.    A  large  wire  conducts  better  than  a  small  one,  and  the 
larger  the  wire  the  less  the  resistance. 

3.  TJie  resistance  varies  directly  as  the  length.     A  wire  one 
mile  long  offers  twice  the  resistance  that  would  be  offered  by  a 
similar  wire  half  a  mile  long. 

The  unit  of  resistance  is  called  an  Ohm,  and  is  very  nearly 
that  of  a  pure  copper  wire,  -fa  inch  in  diameter  and  250  feet  long, 
or  that  of  330  feet  of  ordinary  iron  wire,  -fffy  of  an  inch  in 
diameter. 

The  Siemen's  Unit  is  the  resistance  of  a  column  of  pure  mer- 
cury 1  metre  long  and  of  1  square  millimetre  section,  at  0°  Centi- 
grade, and  is  .9705  British  Association  Units,  or  ohms.  The 


STRENGTH     OP    CURRENT. 


43$ 


resistance  of  ordinary  telegraph  wire  No.  8  is  about  13  ohms  pel- 
mile. 

At  a  mean  temperature  of  20°  C.,  an  increase  of  1°  0.  m 
temperature  produces  an  increased  resistance  in 

Pure  solid  metals  of  about  -fa    of  1  per  cent. 

German  silver  " 

Mercury  " 

The  specific  resistance  of  any  substance  is  determined  for  a 
cube  of  that  substance  measuring  one  centimetre  upon  the  edge. 
Since  resistance  varies  as  length  divided  by  cross-section,  we  can 
determine  the  resistance  of  any  conductor  by  the  formula, 

'  ' 


in  which  R  =  the  whole  resistance,  L  =  length  in  centimetres, 
A  =  area  of  cross-section  in  sq.  cm.,  and  r  =  the  specific  resist- 
ance as  given  in  the  following  table  : 


Specific  Resistance  in  Ohms. 

Silver 000001609 

Copper 000001642 

Iron 000009827 

Lead ,     00001 9850 

German  Silver 000021170 

Mercury 000096190 


Specific  Resistance  in  Ohms. 

Pure  Water  at  22°  C 71 .8 

Dilute  Sul.  Acid,  Tj^ 3.32 

Dilute  Sul.  Acid,  A 1.26 

Glass  at  200°  C 22700000 

Gutta  Percha  at  20°  C. .          35  x  1013 


The  prefixes  mega  and  micro  are  used  to  denote  a  million,  or 
one  millionth  ;  thus,  one  million  ohms  =  one  megohm,  and  one 
millionth  of  a  volt  =  one  microvolt.  These  prefixes  are  applied 
to  the  various  electric  units. 

691.  Strength  of  Current. — Knowing  the  electromotive 
force  and  the  resistances,  we  next  seek  to  determine  the  strength 
of  the  current,  or  the  quantity  which  flows  through  the  conductor 
in  a  given  time. 

The  practical  unit  of  strength  of  current  is  called  one  "  Weber 
per  second"  or  more  commonly  one  "  Weber"  and  is  that  which 
flows,  in  one  second,  through  a  resistance  of  one  ohm  under  an 
electromotive  force  of  one  volt.  It  has  been  proposed  that  this 
unit  of  current,  in  future,  shall  be  called  an  Ampere. 

692.  Ohm's  Law. — It  is  evident  that  as  the  electromotive 
force  is  made  greater,  the  resistance  being  supposed  unchanged, 
the  quantity  of  electricity  which  flows  per  second  will  be  greater 
also  ;   and  as  the  resistance  is  made  greater,  the  electromotive 
force  being  unchanged,  the  current  will  be  less  per  second.     The 
relations  of  strength  of  current,  resistance  and  electromotive  force 
first  stated  by  Ohm.  and  therefore  called  Ofint's  Law,  are  expressed 

28 


434  DYNAMICAL    ELECTRICITY. 

thus  :  The  strength  of  the  current  varies  directly  as  the  electro- 
motive  force  and  inversely  as  the  resistance. 

Expressing  this  as  a  working  formula  we  have 


in  which  G  =  the  number  of  webers  per  second  of  current,  E  = 
the  number  of  volts  of  electromotive  force,  and  R  =  the  number 
of  ohms  of  resistance  in  the  entire  circuit,  within  the  cells  as  well 
as  in  the  external  conductor. 

693.  Manner  of  Connecting  the  Elements  of  a  Bat- 
tery.— The  electromotive  force  of  cells  connected  "in  series," 
that  is,  having  the  zinc  of  the  first  connected  with  the  copper  of 
the  second,  the  zinc  of  the  second  with  the  copper  of  the  third, 
and  so  on  to  the  last  whose  zinc  is  joined  by  the  external  circuit 
with  the  copper  of  the  first,  is  equal  to  the  sum  of  the  electro- 
motive forces  of  the  cells  taken  separately ;  and,  since  the  current 
traverses  the  cells  in  succession,  the  internal  resistance  is  equal  to 
the  sum  of  the  internal  resistances. 

Let  us  call  the  electromotive  force  of  each  cell  E,  the  external 
resistance  of  the  conducting  wire  R,  and  the  internal  resistance 
of  each  cell,  due  to  the  liquid  which  the  current  must  traverse,  r  ; 
then  in  a  battery  of  n  cells  we  have  strength  of  current, 

C= B,  according  to  Ohm's  Law. 

n  r  +  R 

Suppose  we  have  a  number  of  Daniell  cells  of  electromotive 
force  1  volt  each,  and  of  internal  resistance  of  3  ohms  (Art.  690), 
and  that  we  wish  to  send  a  current  of  -^  of  1  weber  per  second 
through  a  circuit  wire  10  miles  long  and  offering  a  resistance  of 
13  ohms  per  mile  (Art.  690). 

If  we  use  one  cell  we  have, 

1  volt 

0  =  o — r T^TT— i =  T-J-?  weber  per  second. 

3  ohms  +  130  ohms 

If  we  join  two  cells  in  series  we  have, 

~  2  volts  „         , 

C  =  - — = — — — : =  -Jjr  of  a  weber  per  second. 

6  ohms  -f  130  ohms 

If  we  use  ten  cells  we  find, 

10  volts 

C  =  0.    ,       — r^A— r: —  =  -Ar  weber  per  second. 
30  ohms  +  130  ohms 

Had  we  joined  our  cells  in  ' f  multiple  arc,"  that  is,  joined  all 
the  zincs  together  to  form  one  electrode  and  all  the  coppers 
together  for  the  other  electrode,  the  effect  would  have  been  the 
snm°  os  that  produced  by  a  single  cell  having  plates  10  times  the 


COUPLING    CELLS.  435 

area  of  the  plates  of  one  cell  of  the  battery  ;  there  would  be  no 
increase  of  electromotive  force,  but  the  internal  resistance  would 
be  only  ^  that  of  a  single  cell,  or  ^  of  an  ohm;  for  the  current 
now  divides  and  traverses  the  liquid  in  all  the  cells  at  the  same 
time,  instead  of  successively,  and  the  total  cross-section  of  liquid 
is  the  sum  of  the  cross-sections  of  the  single  cells,  and  as  the 
resistance  is  inversely  as  the  cross-section,  we  have  the  above 
result. 

Suppose  we  desire  a  strong  current  through  an  external  circuit 
of  six  ohms  resistance,  and  have  eight  cells  like  those  above  ;  if 
we  connect  them  "  in  series  "  we  have 

Q 

C  =  2  -  8~T~6~  —  A  weber  Per  second. 
If  we  join  them  in  "  multiple  arc  "  we  get 

®  =  a    I    f?  —  A  weber  per  second. 

If  we  now  join  them  in  "multiple  arc"  in  groups  of  two 
each,  and  then  join  the  four  groups  in  "  series  "  we  have 

4 
C  =  -  -  -  --  —  =  -J  weber  per  second, 

•§•    X     4:    ~j-    D 

the  best  combination  of  the  three  which  we  have  tried. 

The  maximum  current  is  obtained  when  the  internal  resistance 
equals  the  external  resistance. 

Let  N=  number  of  cells,  r  =  internal  resistance  of  each  cell, 
R  =  external  resistance  of  the  circuit,  g  =.  number  of  cells  in  a 
group  joined  in  "  multiple  arc,"  and  S  =  number  of  groups  joined 
in  "series  ;  "  then 

C  =  ---  ;  but  N  =  g  x  S,  whence  g  =  -~, 

-8+  R 
9 

O     77T 

therefore  C  =  -^  ---  .     If  now  the  total  internal  resistance, 


•—.  r,  equals  the  external  resistance  R,  we  have 

^      TTl 

C  =  x—  ,5  ,  a  maximum  value. 

A    H 

For,  if  the  value  of  C  be  not  greatest  when  the  internal  resistance 
equals  the  external,  let  us  increase  or  decrease  the  former,  which 
we  may  do  by  increasing  or  decreasing  the  number  of  groups. 

N 

For  S  substitute  S  ±  a  ;  then  g  =  ,  which  values  sub- 

o  ~r  a 


436  DYNAMICAL  ELECTRICITY 

stituted  in  the  formula  for  C  give  Ce  =  7-77^  —  "~0  '    -  —  .     But 

(iS  ±  ayr 

~w  —  +  R 

S2  r  N  R 

-j=-  =  R,  by  our  hypothesis,  and  hence  r  =  -^-  ,  which  values 

give  C'  =  -    —^-  —  ^—  —  5-=;  comparing  this  value  with  the 
*  a  K       a  u 


former  we  have 


' 


n  _  r'  -          _  ± 

~  ~ 


which,  reducing  to  common  denominator  and  cancelling,  gives 
G  -  C'  sg^-gr-^qp^,  a  remainder  which  is  positive  ; 

hence  G  must  be  greater  than  C',  and  therefore  that  combination 
which  makes  the  internal  resistance  equal  to  the  external  gives  the 
greatest  current. 

694.  Galvanic  and  Frictional  Electricity  Compared.  — 
The  electricities  furnished  by  chemical  action  and  by  friction  are 
undoubtedly  the  same  in  kind.  But  they  differ  in  that  the  former 
is  produced  in  greater  quantity,  while  the  latter  is  in  a  state  of 
greater  intensity.  This  will  be  understood  by  referring  to  heat. 
The  quantity  of  heat  in  a  warm  room  is  vastly  greater  than  that 
in  the  flame  of  a  lamp  ;  yet  the  former  is  agreeable,  while  the 
latter,  if  touched,  causes  severe  pain  by  its  greater  intensity.  In 
a  similar  manner,  a  quantity  of  galvanic  electricity  may  pass 
through  the  body  without  harm,  which,  if  it  possessed  the  inten- 
sity of  frictional  electricity,  would  instantly  destroy  life. 

The  word  tension,  or  intensity,  expresses  the  degree  of  force 
exerted  by  electricity  in  overcoming  a  given  obstacle,  as  a  break 
in  a  circuit. 

(1)  From  this  difference  in  quantity  and  intensity  results  a 
very  great  difference  in  continuance  of  action.  This  is  indicated 
by  the  terms  dynamical  and  statical.  Galvanic  electricity,  being 
produced  in  prodigious  quantities  and  with  very  feeble  electro- 
motive force,  may  flow  in  a  steady,  gentle  stream  for  many  hours, 
and  is  hence  called  dynamical,  while  frictional  electricity,  being 
small  in  quantity  and  having  great  electromotive  force,  darts 
through  an  opposing  medium  instantaneously,  and  with  great 
violence.  What  motion  it  has  is  therefore  merely  incidental  to  its 
passage  from  one  state  of  rest  to  another.  Hence  the  propriety 
of  the  term  statical.  The  Holtz  machine  has  been  made  to  give 
a  continuous  current  like  that  of  a  battery  of  cells,  the  strength 


CHARACTERISTICS.  437 

of  the  current  being  nearly  proportional  to  the  velocity  of  rota- 
tion. The  electromotive  force  of  the  machine  was  about  53,000 
volts,  at  all  speeds.  The  resistance,  at  120  revolutions  per  minute, 
was  2,810  megohms  (Art.  690),  and  at  450  revolutions  per  minute 
only  646  megohms. 

(2)  Again,  owing  to  its  low  tension,  galvanic  electricity  will 
traverse  many  thousands  of  feet  of  wire  rather  than  pass  through 
the  thin  covering  of  silk  with  which  the  wire  is  insulated,  and 
which  would  be  but  a  slight  obstacle  in  the  path  of  frictional 
electricity. 

But  by  joining  cells  in  series,  the  current  approaches  more 
nearly  to  the  character  of  the  frictional  current.  A  battery  of 
3,520  zinc  and  copper  cells,  in  series,  gave  a  succession  of  sparks 
between  poles  separated  -^  of  an  inch. 

(3)  Comparisons  have  been  made  of  the  actual  quantities  of 
electricity  obtained  by  chemical  action  and  by  friction.     Faraday 
has  shown  that  to  decompose  one  grain  of  water  into  its  constituent 
elements,  oxygen  and  hydrogen,  requires  an  amount  of  frictional 
electricity  equal  to  the  charge  of  a  Leyden  battery  with  a  metallic 
surface  of  thirty-two  acres,  equal  to  a  very  powerful  flash  of  light- 
ning ;  but  by  a  galvanic  current,  furnished  by  three  or  four  Grove 
cells,  the  same  result  is  accomplished  in  a  few  minutes. 

From  this  some  idea  may  be  formed  of  the  vast  quantity  of 
electricity  produced  during  the  steady  flow  for  several  hours  of  a 
Grove  or  Bunsen  battery. 

The  deflection  of  a  magnetic  needle  (as  explained  hereafter) 
depends  solely  upon  the  quantity  of  the  current  flowing  around  it. 
For  this  reason  the  current  from  a  frictional  or  induction  machine, 
being  of  small  quantity  through  having  great  electromotive  force, 
will  not  deflect  the  needle  unless  special  arrangements  for  multiply- 
ing the  effects  are  made  ;  on  the  other  hand  Faraday  immersed  a 
zinc  and  a  platinum  wire,  each  ^  inch  in  diameter,  in  acidulated 
water  to  the  depth  of  -|  of  an  inch,  during  ^j-  of  a  second,  and  the 
current  thus  generated  produced  a  greater  effect  upon  the  needle 
thaA  was  produced  by  28  turns  of  the  large  electric  machine  of 
the  Royal  Institution. 


438 


DYNAMICAL    ELECTRICITY. 


FIG.  375. 


CHAPTER    II. 

ELECTRO-MAGNETISM. 

695.  Mutual  Action  of  Currents.— 

1.  If  galvanic  currents  flow  through  parallel  wires  in  the  same 
direction,  they  attract  each  other ;  if  in  opposite  directions,  they 
repel  each  other.     These  effects  are  shown  by  suspending  wires, 
bent  as  in  Fig.  375,  so  that  their  lower  ends  may  dip  into  four 
separate  mercury  cups  a,  b,  a',  V,  by  means  of  which  connection 

between  the  wires  G  and  D 
and  the  battery  may  be 
readily  made.  The  sus- 
pending threads  should  be 
two  or  three  feet  long,  and 
the  mercury  cups  should  be 
large  enough  to  allow  con- 
siderable lateral  movement 
of  the  wires.  If  simultane- 
ous currents  be  sent  through 
the  two  wires  G  and  D,  in 
the  same  direction,  the  wires  will  move  towards  each  other;  if 
currents  be  sent  through  the  wires  in  opposite  directions  at  the 
same  time,  they  will  separate  more  widely. 

Hence,  when  a  current  flows  through  a  loose  and  flexible 
helix,  each  turn  of  the  coil  attracts  the  next,  since  the  current 
moves  in  the  same  direction  through  them  all.  In  this  way,  a 
coil  suspended  above  a  cup  of  mercury,  so  as  to  just  dip  into  the 
fluid,  will  vibrate  up  and  down  as  long  as  a  current  is  supplied. 
The  weight  of  the  helix  causes  its  extremity  to  dip  into  the 
mercury  below  it ;  this  closes  the  circuit,  the  current  flows  through 
it,  the  spirals  attract  each  other,  and  lift  the  end  out  of  the  mer- 
cury ;  this  breaks  the  circuit,  and  it  falls  again,  and  thus  the 
movement  is  continued. 

2.  If  currents  flow  through  two  wires  near  each  other,  which 
are  free  to  change  their  directions,  the  wires  tend  to  become  paral- 
lel to  each  other,  with  the  currents  flowing  in  the  same  direction. 
Thus,  two  circular  wires,  free  to  revolve  about  vertical  axes,  when 
currents  flow  through  them,  place  themselves  by  mutual  attrac- 


CURRENTS    NOT     PARALLEL. 


439 


tions  in  parallel  planes,  as  in  Fig.  376,  or  in  the  same  plane,  as 
in  Fig.  377.  In  the  latter  case,  we  must  consider  the  parts  of  the 
two  circuits  which  are  nearest  to  each  other  as  small  portions  of 
the  dotted  straight  lines,  c  d  and  ef 


FIG.  377. 


d    £ 

It  appears,  therefore,  that  galvanic  currents,  by  mutual  attrac- 
tions and  repulsions,  tend  to  place  themselves  parallel  to  each  other 
in  such  a  manner  that  thefloio  is  in  the  same  direction. 

696.  Currents  not  Parallel.— Currents,  both  of  which  flow 
towards  a  common  point,  or  both  of  which  flow  away  from  a  com- 
mon point,  attract  each  other. 

If  one  of  two  currents  flows  towards,  and  the  other  away  from 
a  common  point,  the  two  currents  repel  each  oilier. 

These  cases  are  evident  deductions  from  the  preceding  para- 
graph. Suppose  the  two  currents  (Fig.  378)  to  t  flow  in  A  and  B 
as  though  they  came  from  C,  then  the  tendency  of  the  wires  A 
and  B  is  towards  parallelism, 

T?Tn     QTQ 

and  as  we  suppose  the  cur- 
rents to  flow  from  the  direc- 
tion C,  the  wires  must  tend 
to  move  towards  each  other 
in  order  to  become  parallel. 
The  same  effect  would  be 

produced  if  the  currents  in  A  and  B  were  to  flow  towards  C. 
But  if  the  current  in  A  flows  from  the  direction  C,  and  that  in  B 
towards  the  point  C,  then  the  tendency  of  the  wires  to  become 
parallel,  with  the  currents  flowing  in  the  same  direction,  causes  B 
to  revolve  about  (7  as  a  centre  till  it  reaches  the  position  B',  and 
then  the  condition  that  the  currents  shall  flow  in  the  same  direc- 
tion will  be  fulfilled.  It  is  not  necessary  that  we  should  regard 
A  and  B  as  lying  in  the  same  plane. 

697.  Continuous  Rotation   Produced  by  Mutual  Ac- 


440 


DYNAMICAL    ELECTRICITY. 


FIG.  379. 


tion  of  Currents. — Suppose  a  continuous  current  to  flow 
through  a  wire  A,  as  indicated  in  Fig.  379,  and  that  a  wire  B,  so 
bent  as  to  dip  into  the  mercury  cup  m  at  one  end,  and  into  the 

annular  mercury  trough  n 
at  the  other,  be  suspended 
at  the  middle,  a  counter- 
poise (7  keeping  it  balanced. 
If  now  a  current  be 
made  to  flow  from  the  cup 
m,  through  B,  and  thence 
out  again  by  means  of  the 
mercury  contact  in  n,  the 
wire  B  will  rotate  in  a  direc- 
tion opposite  to  that  of  the 
current  in  A  ;  for  the  cur- 
rent in  B,  and  that  in  the  part  of  A  to  the  right  of  n,  are  both 
flowing  towards  n  and  hence  attract,  while  the  current  in  B  and 
that  part  of  the  current  in  A  immediately  to  the  left  of  n  are 
flowing  in  directions  to  cause  repulsion. 

A  sinuous  current  produces  the  same  effect  as  a  straight  cur- 
rent having  the  same  general  direction  and  length.  If  a  con- 
ductor, having  one  portion  sinuous  and  the  other  straight,  be 

FIG.  380. 


FIG.  381. 


bent  as  in  Fig.  380,  so  that  the  current  may  flow  from  a  to  b 
through  the  straight  part,  and  from  b  to  c  through  the  sinuous 
part,  the  two  portions  of  the  current  thus  flowing  close  together 
in  opposite  directions,  the  joint  effect  upon  a  movable  conductor 
parallel  to  a  b  will  be  inappreciable. 

698.  Helices. — A  wire  bent  in  a  spiral,  as  in  Fig.  381,  is 
called  a  coil  or  helix.  If  the  wire  is  coiled  in  the  direction  of  the 
thread  of  a  common  or 
right-hand  screw  (Art.  134), 
it  is  called  a  right-hand 
helix;  if  in  the  direction 
of  the  thread  of  a  left-hand 
screw,  it  is  called  a  left-hand 
.helix.  Without  referring  to 
the  screw,  the  distinction 
between  the  right  and  left 
hand  helix  may  be  described  thus  :  When  a  person  looks  at  a  helix 
in  the  direction  of  its  length,  if  the  wire,  as  it  is  traced  from  him, 


FIG.  382. 
Jeff, 


THESOLENOID.  441 

winds  from  the  left  over  to  the  right,  it  is  a  right-hand  helix  (Fig. 
381) ;  if  from  the  right  over  to  the  left,  a  left-hand  helix  (Fig. 
382). 

699.  The  Solenoid. — Let  a  helix  be  constructed  as  in  Fig. 
383,  in  which  the  ends  are  turned  back  through  the  coil,  metallic 
contact  being  avoided  through- 
out; this  is  called  a  solenoid— 

that  is,  a  tubular  or  channel- 
shaped  magnet.  Next,  let  the 
electrodes  p  and  n  of  a  battery 
be  furnished  with  sockets,  one 
vertically  above  the  other,  in 
which  the  two  ends  of  the  helix 
wire  are  placed.  The  solenoid 
is  then  free  to  turn  nearly  a  whole  revolution  around  a  vertical 
axis,  at  the  same  time  that  a  current  is  passing  through  it.  The 
helix  is  supposed  to  be  a  left-hand  one,  and  is  so  connected  with 
the  battery  that  the  current  passes  through  it  from  N  to  $,  and 
therefore  around  it  from  right  over  to  left. 

The  direction  of  the  spirally  coiled  wire,  at  any  point,  may  be 
resolved  into  two  other  directions,  one  parallel  to  the  axis  of  the 
helix,  and  the  other  in  a  plane  perpendicular  to  this  axis;  the 
effect  of  the  former  components  is  equal  and  contrary  to  that  of 
the  straight  returned  portion  N  S,  and  is  thus  neutralized,  while 
the  latter  components  act  as  a  series  of  circular  parallel  currents 
in  planes  at  right  angles  to  the  axis. 

While  the  current  flows,  the  following  phenomena  may  be 
observed  : 

1.  If  a  magnet  be  brought  near  it,  JVwill  be  attracted  by  the 
south  pole,  and  S  by  the  north  pole.     If,  instead  of  a  magnet, 
another  solenoid  be  presented  to  it,  whose  corresponding  extremi- 
ties are  N1  and  S',  N  and  S1  will  attract  each  other,  as  also  S 
and  N'. 

2.  If  not  disturbed,  the  coil  will  place  itself  lengthwise  in  the 
direction  of  the  magnetic  meridian,  with  the  extremity  N  toward 
the  north,  and  S  toward  the  south. 

3.  If  a  bar  of  iron  be  placed  within  it,  the  bar  will  become  a 
magnet,  having  its  north  pole  at  N,  and  its  south  pole  at  S. 

If  a  right-hand  helix  had  been  employed,  all  these  phenomena 
would  have  been  reversed. 

700.  Ampere's  Theory  of  Magnetism. — In  these  experi- 
ments a  coil  is  found  to  act  the  same  as  a  magnet  whose  north  and 
south  poles  are  at  JV^and  8  respectively.     We  therefore  deduce  the 
following : 


442 


DYNAMICAL    ELECTRICITY. 


FIG.  385. 


1.  A  helix  traversed  by  a  galvanic  current  is  a  magnet  the 
position  of  whose  poles  depends  on  the  direction  of  the  current. 

2.  Conversely,  a  magnet,  like  a  coil,  may  be  conceived  to  owe 
its  magnetic  properties  to  currents  of  electricity  which  traverse  it. 

This  is  the  theory  of  Ampere,  and  is  the  one  generally  received, 
notwithstanding  some  objections  to  it. 

In  the  helix  a  single  current  is  present.     But  in  a  magnet  we 
must  conceive  of  an  infinite  number  of  currents,  the  circuit  of 
each  being  confined  to  an  individual  molecule.    Fig.  384  repre- 
sents a  magnet  accord- 

FIG.  384.  ing  to  this  theory,  and 

N  and  S  (Fig.  385) 
show  the  extremities 
of  the  north  and  south 
poles  on  a  larger  scale.  The  arrows  on  the  convex  surface  show 
the  general  direction  of  all  the  currents — that  is,  of  those  portions 
of  them  nearest  the  surface, 
where  magnetism  is  in  fact 
developed — and  may  there- 
fore represent  them  all. 

Since  /Sis  the  south  pole 
of  the  magnet,  as  supposed 
to  be  seen  by  an  observer 
looking  at  it  in  the  direc- 
tion of  its  axis,  it  follows 
that  when  a  magnet  is  in 
its  normal  position,  that  is, 
with  its  north  pole  pointing 

northward,  its  currents  circulate  from  west  over  to  east,  and  there- 
fore from  left  over  to  right  if  the  observer  is  also  looking  north- 
ward. 

These  supposed  currents  of  the  magnet  are  so  small  that  we 
cannot  take  cognizance  of  them  directly.  But  on  the  basis  of 
Ampere's  theory,  we  may  substitute  for  them  the  large  and  man- 
ageable current  of  a  helix.  Then,  by  determining  experimentally 
the  causes  of  magnetic  phenomena  in  the  case  of  the  latter,  we 
may  assign  the  same  causes  to  like  phenomena  of  the  magnet. 

701.  Relations  of  Currents  and  Magnets  to  Each 
Other. — It  should  be  constantly  borne  in  mind  that  when  we 
hold  a  magnet  before  us,  the  north  seeking  end  farthest  from 
us,  so  that  we  are  looking  along  its  length  from  S  to  N9  the  cur- 
rents circulate  around  the  magnet  from  left  over  to  right,  or  in 
the  direction  in  which  ive  would  turn  an  ordinary  screw  when, 
driving  it  into  wood. 


CURRENTS    AND    MAGNETS. 


443 


1.  When  two  solenoids,  suspended  as  in  Fig.  383,  or  when  a 
solenoid  and  a  magnet,  or  two  magnets,  are  brought  near  each 
other,  poles  of  different  names  attract,  and  those  of  the  same  name 
repel.  For,  when  the  magnets  suspended  from  A  and  B  (Fig.  386) 

FIG.  386. 


are  in  the  same  line,  it  is  seen  that  the  currents  are  parallel  and 
flow  in  the  same  direction  in  all  the  corresponding  parts ;  and  in 
Fig.  387,  where  they  hang  side  by  side,  the  nearer  parts  of  the 

FIG.  387.  FIG.  388. 

A\ 


currents  are  parallel  and  flow  in  the  same  direction.  While  in 
Fig.  388,  where  like  poles  are  contiguous,  the  corresponding  parts 
of  the  currents  flow  in  opposite  directions. 

2.  When  a  magnet  is  suspended  within  a  loop  through  which 
a  current  flows,  if  free  to  move  FIG<  339. 

it  will  place  itself  at  right  angles 
to  the  plane  of  the  circuit,  with 
the  north  pole  pointing  toward  a 
person,  when  the  current  passes 
from  his  right  over  to  his  left 
(Fig.  389).  Therefore  if  the  cir- 
cuit is  in  a  horizontal  plane, 
the  magnet  turns  its  north 
pole  downward,  if  the  current 
flows  as  in  Fig.  390,  or  upward  if  the  current  is  reversed. 


444 


DYNAMICAL    ELECTRICITY 


3.  When  a  magnet  is   brought  near  a  closed   circuit  wire, 
as  H (Fig.  391),  it  will  place  itself  tangentially  to  a  circle,  x  y  z, 


FIG.  390. 


whose  centre  is  in  the  wire,  and  its  plane  perpendicular  to  it. 
The  part  of  the  wire  nearest  to  the  magnet  may  be  considered  as 
a  small  portion  of  a  loop  around  it,  as  in  Fig.  389.  This  tangen- 
tial relation  is  maintained  on  all  sides  of  the  circuit,  it  being 
everywhere  true  that  when  the  south  pole  is  directed  to  a  person, 
the  current  descends  on  the  right,  as  if  it  had  passed  from  the  left 
over  to  the  right. 

Comparing  Figs.  390  and  391,  it  is  evident  that  the  current 
and  the  magnet  may  change  places  without  disturbing  their  rela- 
tive directions,  it  being  understood  that  the  current  flotus  in  the 
same  direction  in  which  the  north  pole  points. 

702.  Magnet  Used  to  Measure  Current. — If  a  mounted 
magnetized  needle  be  allowed  to  come  to  rest  it  will  place  itself  in 
the  magnetic  meridian  with  its  marked  end  toward  the  north  as 
in  Fig.  392.  If  a  wire  be  held  above  it  and  parallel  to  it,  (Fig. 
392),  and  a  current  be  sent  through  the  wire  from  A  to  B,  the 


FIG.  392. 


FIG.  393. 


needle  will  be  deflected,  the  marked  end  moving  to  the  west 
(Art.  701).  If  the  current  be  passed  Mow  the  needle  from  South 
to  North,  the  marked  end  would  move  to  the  East.  If  a  con- 


GALVANOMETER.  445 

tinuous  current  be  sent  around  the  needle  in  the  plane  of  the 
magnetic  meridian  (Fig  393),  from  South  to  North  above  and 
from  North  to  South  below  the  needle,  each  of  the  two  branches 
of  the  encircling  wire  tends  to  move  the  marked  end  of  the  needle 
to  the  West,  and  the  two  together  produce  double  the  deflection 
due  to  either  alone.  The  discovery  of  these  effects  was  made  by 
Oersted  in  1820. 

703.  Galvanometer. — The  action  of  the  current  upon  the 
needle  is  proportional  to  the  quantity  of  electricity  which  floivs 
around  it  in  a  unit  of  time. 

When  the  coil  consists  of  many  convolutions  of  wire,  a  very 
feeble  current  passing  through  will  deflect  the  needle  from  its 
north  and  south  direction,  and  the  amount  of  deflection,  not 
exceeding  20°,  serves  as  a  measure  of  the  galvanic  force.  Hence 
the  name  of  the  instrument.  To  render  it  still  more  sensitive,  a 
second  smaller  needle,  with  poles  reversed,  attached  to  the  same 
vertical  wire,  makes  the  first  nearly  astatic  with  relation  to  the 
earth.  In  making  such  a  coil,  the  wire 
must  be  carefully  insulated.  This  is  gen- 
erally done  by  winding  it  with  silk  thread. 
In  Fig.  394  the  galvanometer  is  repre- 
sented as  covered  by  a  bell-glass.  The 
coil  is  seen  beneath  the  graduated  circle. 
The  two  needles  of  the  astatic  combina 
tion  are  shown,  the  first  above  the  gradu- 
ated scale  and  the  other  just  protruding 
from  the  coil,  beneath  the  first,  and  paral- 
lel to  it.  The  character  of  the  coil  should  depend  upon  the  cur- 
rent to  be  measured  ;  for  currents  which  have  already  traversed 
great  resistances,  many  hundred  turns  of  fine  wire  are  needed, 
while  for  thermo-electric  currents  (Art.  706)  a  few  turns  of  thick 
wire  are  most  suitable.  By  means  of  properly  insulated  coils  of 
fine  wire,  of  3000  or  more  turns,  the  current  of  the  Holtz  machine 
may  be  made  to  produce  an  effect  upon  the  needle. 

704.  Sine  Galvanometer. — It  was  stated,  Art.  703,  that  up 
to  about  20°  the  deflections  of  the  needle  were  proportional  to  the 

strength  of  the  current,  the  conducting 
wire  remaining  in  the  plane  of   the 
^          magnetic  meridian.    Now  suppose  that 
f,s' t  the  conducting  wire  be  made  to  follow 

JL  y''s''          _B      the  needle,  always  remaining  parallel 

to  it,  as  in  Fig.  395,  in  which  the  wire 
A  B  has  followed  the  needle,  till  a 
position  of  rest  N'  $'  has  been  found, 


446  DYNAMICAL    ELECTRICITY. 

in  which  position  the  deflecting  force  of  the  current  in  B'  A'  just 
balances  the  directive  force  of  the  earth's  magnetism. 

Let  H  K  (Fig.  396)  represent  the  magnetic  meridian,  and  B  A 
the  direction  of  the  current,  parallel  to  and  above  the  needle.  Let 
Nm  represent,  in  intensity  and  direction, 
the  directive  force  of  the  earth's  magnetism 
at  the  place.  Now  N  m  may  be  resolved 
into  two  components ;  o  w,  in  the  direction 
of  the  needle,  has  no  tendency  to  produce 
deflection,  while  No,  at  right  angles  to 
the  needle,  equals  and  opposes  the  devia- 
ting effect,  represented  by  N  o',  of  the 
cnrrent  in  B  A.  From  the  triangle  N  o  m 
JT  we  have  N  o  =  N  m  x  sine  N  m  o  = 

Nm  x  sine  N  C  H.  Hence,  since  the  factor  Nm  is  constant,  it 
being  the  directive  force  of  the  earth's  magnetism  at  the  place, 
the  deviating  effect  of  the  current  is  proportional  to  the  sine  of 
the  angle  of  deflection.  Any  sensitive  galvanometer  may  be  used 
as  a  Sine  Galvanometer  by  turning  the  coil,  while  the  current  is 
flowing,  till  it  is  parallel  to  the  needle  in  its  position  of  rest ;  if 
now  the  circuit  be  broken,  the  needle  will  return  to  the  plane  of 
the  magnetic  meridian,  and  the  sine  of  the  number  of  degrees 
passed  over  will  be  the  relative  strength  of  the  current. 

If  the  coil  be  of  great  diameter  as  compared  with  the  length  of 
the  inclosed  needle,  the  strength  of  the  current  is  proportional  to 
the  Tangent  of  the  angle  of  deflection,  the  plane  of  the  coil  being 
coincident  with  the  plane  of  the  magnetic  meridian.  An  instru- 
ment the  diameter  of  whose  coil  is  ten  to  fifteen  times  the  length 
of  the  included  needle  is  called  a  Tangent  Galvanometer. 

705.  Polarity  with  Respect  to  the  Earth.— It  is  believed 
that  currents  of  electricity  are  constantly  traversing  the  earth's 
crust,  passing  around  it  from  east  to  west,  and  making  the  earth 
itself  a  magnet,  with  boreal  magnetism  developed  at  the  north 
pole,  and  austral  at  the  south  pole.  Thus  the  earth  may  be  taken 
as  the  standard  magnet,  and  both  it  and  the  currents  around  it 
control  the  polarity  of  the  needle.  For,  as  in  Fig.  397,  in  order 
that  the  current  of  the  magnet  may  be  parallel  with  the  adjacent 
terrestrial  current,  and  in  the  same  direction  with  it,  since  the 
latter  passes  from  east  to  west,  the  lower  side  of  the  former  must 
also  pass  from  east  to  west.  But  in  order  that  this  may  be  the 
case,  the  north  pole  of  the  magnet  must  point  northward,  and 
this  it  does  when  free  to  obey  the  directive  influence  of  the  earth. 

At  first  view,  the  earth  currents  from  east  to  west  seem  to  be 
in  the  wrong  direction  ;  for  that  is  from  left  over  to  right,  to  a 


T  H  E  R  M  0-E  L  E  C  T  R  I  0  I  T  Y . 


447 


person  to  whom  the  north  pole  points.  This,  however,  is  ex- 
plained by  recollecting  that  the  magnetism  of  the  north  pole  of 
the  earth  is  the  same  as  that  of  the  south  pole  of  a  magnet  (Art. 


FIG.  397. 


595).  For  convenience,  that  end  of  a  needle  which  points  north 
is  called  the  north  pole ;  but  by  the  law  of  attraction  between 
opposite  poles,  it  must  be  unlike  the  north  pole  of  the  earth. 
Therefore,  the  rule  for  the  direction  of  currents  around  a  magnet 
must  be  reversed  when  applied  to  the  earth. 

The  existence  of  currents  traversing  the  earth's  crust  has  been 
variously  accounted  for.  The  strong  analogy  between  them  and 
those  of  thermo-electricity  points  to  the  heat  of  the  sun  as  at  least 
a  very  probable  cause. 

706.  Thermo-Electricity. — Let  two  bars  of  bismuth  (#) 
and  antimony  (a)  be  soldered  together  as  in  Fig.  398.  If  now  the 

FIG.  398. 


joint  s  be  heated  by  a  lamp  a  current  will  flow  across  the  heated 
junction  from  the  bismuth  to  the  antimony,  as  will  be  shown  by 
the  galvanometer  G. 

The  electromotive  force  of  the  current  depends  upon  the 
metals  in  contact  at  the  heated  junction.  If  any  one  of  the 
metals  given  below  be  joined  with  any  one  following  it  in  the  list, 
upon  applying  heat  the  current  will  flow  across  the  junction  from 
the  former  to  the  latter:  Bismuth,  platinum,  lead,  tin,  copper, 


448 


DYNAMICAL    ELECTRICITY. 


silver,  zinc,  iron,  antimony.     In  some  cases  a  continued  increase 
of  temperature  at  a  junction  finally  reverses  the  current. 

707.  Thermo-Electric  Pile. — If  a  series  of  bars  of  bismuth 
and  antimony  be  arranged,  as  in  Fig.   399,  and  the  junctions 
marked  3  and  4  be  equally  heated,  no  current  will  be  indicated  by 

the  galvanometer ;  for  the  flow  at 
3  would  be  from  the  bismuth  to 
the  antimony  as  indicated  by  the 
arrow,  while  at  4  it  would  also  be 
from  b  to  a,  as  shown,  and  these 
two  currents  would  neutralize  each 
other.  But  if  we  heat  only  one  set 
of  junctions,  the  odd-numbered  for 
instance,  then  a  current  flows  whose  electromotive  force  is  pro- 
portional to  the  number  of  heated  junctions. 

A  set  of  twenty  or  thirty  pairs,  conveniently  arranged  so  that 
the  alternate  junctions  may  be  simultaneously  subjected  to  heat- 
ing or  cooling  effects,  is  called  a  thermo-pile,  and  has  been  an 
important  instrument  in  investigations  upon  heat.  The  electro- 
motive force  of  the  best  thermo-electric  piles  is  comparatively  very 
feeble. 

We  may  suppose  that  the  terrestrial  current  may  be  caused,  in 
part  at  least,  by  the  unequal  heating  of  the  heterogeneous  sub- 
stances composing  the  earth's  crust,  as  the  sun's  heat  is  alternately 
poured  upon  and  withdrawn  from  them  once  in  every  diurnal 
revolution. 

708.  Magnetic    Induction    by  Currents.  —  Ampere  ac- 
counted for  the  phenomena  of  magnetic  induction  by  supposing 
that  galvanic   currents   circulate   through   the  molecules  of  all 
bodies,  but  in  different  directions,  so  that  they  mutually  neutralize 
each  other.    That  in  a  few  substances,  such  as  steel  and  iron,  it  is 
possible  to  control  these  currents  and  cause  them  all  to  flow  in 
the  same  direction ;  and  that  when  this  is  done,  the  phenomena 
of  polarity  ensue. 

Supposing  this  to  be  the  correct  explanation,  the  effect  of  a 
galvanic  current  (and  in  fact  of  any  method  of  magnetizing)  is 
simply,  by  repulsion  and  attraction,  to  produce  uniformity  of  direc- 
tion among  these  magnetic  currents. 

709.  The  Permanent  and  Temporary  Magnet— When 
a  current  of  sufficient  strength  is  passed  around  a  bar  of  well- 
tempered  steel,  a  permanent  magnet  of  considerable  power  may 
be  obtained. 

With  soft  iron,  the  result  is  &  temporary  magnet,  which  retains 


THE    U-MAGNET. 


449 


FIG.  400. 


its  magnetic  properties  only  while  the  current  is  in  motion.  In 
either  case  the  poles  are  always  in  the  position  which  those  of  a 
needle  would  voluntarily  assume  if  placed  in  the  same  relation  to 
the  current. 

710.  The  U-Magnet. — Let  a  piece  of  soft  iron,  in  the  form 
of  a  horseshoe  or  the  letter  U  (Fig.  400),  be  wound  with  a  coil  of 
insulated  copper  wire  whose  extremities,  W  and  w9  are  dipped  in 
cups  of  mercury,  in  which  are  also  dipped  the  electrodes  +  and  — 
of  a  battery.    When  all  the  wires  are 

in  metallic  communication,  the  cir- 
cuit is  closed,  and  the  current  pass- 
ing around  the  iron  makes  it  a  mag- 
net ;  and  since  to  a  person  looking 
along  the  length  of  the  helix  the 
current  passes  from  right  over  to 
left,  the  north  pole  is  at  N9  and  the 
south  pole  at  8.  As  soon  as  the 
circuit  is  broken  by  lifting  out  of 
the  mercury  any  one  of  the  wires, 
the  weight  which  was  previously 
sustained  will  fall,  showing  that  the 
iron  is  no  longer  a  magnet. 

711.  Helices. — The  form  of  coil  or  helix  generally  employed 
is  shown  in  Fig.  401.     Many  hundreds  or  even  thousands  of  feet 
of  insulated  wire  are  wound  around  two  bobbins,  and  through  the 
centre  of  each  passes  a  branch  of  the  U-shaped  iron  ;   or,  more 

frequently  the  central  cores  of  iron  are 
separate  pieces,  joined  by  a  third  one 
across  two  of  the  ends,  and  thus  a  U- 
magnet  of  modified  form  is  obtained. 
By  employing  a  fine  wire  coiled  many 
times  around  the  bobbins,  a  magnet  of 
very  great  power  may  be  formed,  con- 
sidering the  weakness  of  the  battery 
which  furnishes  the  current.  A  magnet 
formed  by  the  use  of  a  small  Bunsen 
cell  has  been  known  to  lift  five  hundred 
pounds,  and  with  twenty  Grove  cells  can 
be  made  to  sustain  a  weight  of  three 
tons. 

The  strength  of  the  magnet  increases 
nearly  with  the  strength  of  the  current, 
and  with  the  number  of  coils  of  wire, 

provided  the  diameter  of  the  bobbin  does  not  become  so  great 
29 


FIG.  401. 


450 


DYNAMICAL    ELECTRICITY. 


as  to  remove  the  outer  coils  too  far  for  their  due  effect.  This 
law  is  general,  but  is  true  only  within  certain  limits,  deter- 
mined for  each  bar.  If  the  wire  be  increased  in  length,  the 
resistance  is  also  increased,  and  consequently  less  current  flows 
than  before,  so  that  there  is  a  certain  relation  between  the  length 
of  the  coil  and  the  electromotive  force  which  will  give  the  maxi- 
mum magnetic  effect.  The  rule  in  practical  telegraphy  is  to  make 
the  resistance  of  the  wire  on  the  bobbin  equal  to  that  of  the  cir- 
cuit including  that  of  the  battery  itself. 


CHAPTER    III. 

INDUCED    CURRENTS. 

I.  CURRENTS  INDUCED  BY  CURRENTS. 

712.  Experiments. — Arrange  a  straight  portion  of  a  broken 
galvanic  circuit,  p  n  (Fig.  402),  parallel  to  a  part  b  a  of  a  galva- 
nometer circuit.  Suppose  the  needle  of  the  galvanometer  to  stand 
at  zero.  If  now  the  battery  circuit  be  closed,  by  dipping  the  wires 
into  the  mercury  cup  m,  at  the  instant  of  closing  the  needle  will 

PIG.  402. 


be  deflected  to  that  side  which  indicates  a  current  flowing  in  the 
galvanometer  circuit  in  a  direction  opposite  to  that  of  the  battpry 
current.  This  deflection  of  the  needle  is  but  momentary,  and  it 
returns  speedily  to  zero,  and  remains  at  rest,  although  the  battery 
circuit  remains  closed.  At  the  instant  of  breaking  the  circuit, 
the  needle  is  again  deflected,  but  in  a  direction  indicating  a  flow 
of  current  in  the  galvanometer  circuit  in  the  same  direction  as  that 
of  the  battery.  After  a  few  oscillations  the  needle  comes  to  rest 
at  zero  again. 


INDUCED    CURRENTS.  451 

These  two  currents  in  a  b  are  called  induced  currents  ;  and  the 
one  in  p  n,  to  which  they  owe  their  origin,  is  called  the  inducing 
current.  The  presence,  direction,  and  duration  of  the  induced 
currents  are  indicated  by  the  galvanometer  #. 

713.  Characteristics  of  Induced  Currents. — It  is  obvious 
that  induced  currents  differ  materially  from  the  current  of  a  bat- 
tery which  is  uniform  in  direction  and  constant  in  intensity  for 
an  appreciable  length  of  time.     The  following  are  the  distinctive 
features  of  induced  currents  : 

(1)  Induced  currents  are  instantaneous. 

(2)  On  closing  the  circuit,  the  direction  of  the  resulting  in- 
duced current  is  opposite  to  that  of  the  inducing  current. 

(3)  On  breaking  the  circuit,  the  induced  and  inducing  currents 
are  in  the  same  direction. 

714.  Inducing  and  Induced  Currents  in  one  Wire. — 
We  have  thus  far  considered  only  the  inductive  influence  of  the 
current  on  a  wire  exterior  to  its  circuit. 

In  order  to  produce  the  preceding  results  with  a  single  wire, 
let  the  circuit- wire  be  coiled  as  in  Fig.  403.     Each  spire  is  now 
acted  Upon  inductively  by  the  galvanic  current  passing  through 
the  adjacent  spires  in  the  man- 
ner already  described  for  sepa-  ^IG-  403. 
rate  wires. 

The  result  of  these  several 
inductive  actions  is  that  when 
the  circuit  is  closed  and  broken, 
regular  induced  currents  are 
generated  in  it.  And  since 
these  coexist  for  an  instant  of 
time  with  the  inducing  current, 
and  pass  through  the  same  electrodes  with  it,  it  follows — 

(1)  That  when  the  circuit  is  closed,  the  inducing  current  is 
partially  neutralized,  and  has  its  intensity  diminished  by  the  in- 
duced current  which  flows  in  a  direction  contrary  to  its  own ; 
and 

(2)  That  when  it  is  opened,  the  induced  current  having  now 
the  same  direction  as  the  inducing  current,  reinforces  it  and  aug- 
ments its  intensity. 

715.  Mode  of  Naming  Circuits  and  Currents. — The  phe- 
nomena of  induced  currents  were  discovered  by  Faraday  in  1832, 
and  to  him  we  owe  the  foregoing  description  of  them.    The  fol- 
lowing terms  now  in  use  were  also  introduced  by  him  : 


452  DYNAMICAL    ELECTRICITY. 

The  inducing  current  is  called  the  primary  current,  and  the 
wire  it  traverses  the  primary  wire.  Currents  induced  in  the  pri- 
mary wire  are  called  extra  currents  ;  the  one  obtained  on  closing 
the  circuit  is  the  inverse  extra  current ;  the  one  on  opening  it  is 
the  direct  extra  current  (Art.  713). 

A  wire  exterior  to  the  primary,  as  a  b  in  Fig.  402,  is  ;i 
secondary  wire,  and  the  currents  induced  in  it  are  secondary 
currents. 

716.  Currents  Induced  in  Coils. — Instead  of  straight 
wires  or  loose  spirals,  compact  coils  of  carefully  insulated  wire  are 
employed.  Thus  all  parts  of  the  wire  are  brought  much  nearer 
to  each  other,  and  the  inductive  influence  is  far  more  energetic. 
Indeed,  without  a  coil,  the  presence  of  induced  currents  can 
generally  be  detected  only  with  a  delicate  galvanometer.  The 
following  experiments  show  the  effects  of  coils  : 

(1)  Around  a  hollow  wooden  bobbin,  b  (Fig.  404),  coil  about 
100  feet  of  No.  16  insulated  copper  wire.     Let  this  be  made  a  part 

of  the  circuit  of  a  battery,  as  shown  in  the 
figure.  This  circuit  is  of  course  closed 
when  m  and  n  touch  each  other.  Now  if 
m  and  n  be  held  in  contact,  one  in  each 
hand,  and  then  be  separated,  the  body  of 
the  operator  becomes  a  part  of  the  circuit, 
and  the  primary  current,  not  having  suffi- 
cient intensity  to  pass  through  it,  ceases. 
But  the  direct  extra  current  passes  through,  producing  a  shock. 
When  the  wires  are  brought  together  again,  the  primary  and  inverse 
extra  currents  pass  through  the  metallic  circuit,  and  no  shock  is 
felt. 

The  more  rapid  the  rate  at  which  m  and  n  are  brought  to- 
gether and  separated,  the  more  decided  are  the  results  obtained. 
To  produce  the  most  marked  effect,  attach  a  coarse  file  to  one 
end,  as  m,  and  hold  it  in  one  hand  while  n  is 
drawn  rapidly  over  the  ridges  of  its  surface  with 
the  other. 

(2)  Fig.  405  represents  the  same  coil  as  Fig. 
404,  with  the  addition  of  a  bundle  of  soft  iron 
wires,  w,  inserted  in  the  hollow  bobbin. 

When  the  circuit  is  closed,  these  wires  are 
magnetized — that  is,  the  Amperean  currents  sup- 
posed to  reside  in  them  are  made  to  circulate  in  the 
same  direction  as  the  battery  current  (Art.  708). 

And  since  the  appearance  and  disappearance  of  these  magnetic 
currents  are  simultaneous  with  the  appearance  and  disappearance 


RUHMKOKFF'S    COIL. 


453 


FIG.  406. 


of  the  primary  current,  they  augment  the  effects  of  the  latter, 

and  the  resulting  extra  currents  are  of  greater  intensity. 

The  effect  of  soft  iron  in  the  primary  coil  is  an  observed  fact, 

and  the  above  is  the  way  in  which  Faraday  accounted  for  it  on 

the  basis  of  Ampere's  theory. 

(3)  Let  the  primary  coil  and  bundle  of  wires  of  the  preceding 

figure  be  placed  within  a  secondary  coil,  d  (Fig.  406),  from  which 

it  is  carefully  insulated.    This 

secondary  coil  should  be  made 

of  wire  much  greater  in  length 

and  smaller  in  diameter  than 

that  of  which  the  primary  coil 

is  made.     For  instance,  let  it 

consist  of  1500  feet  of  No.  35 

insulated  copper  wire.     When 

the  ends  of  this  wire,  h,  h',  are 

held  one  in  each  hand,  every 

time   the   primary  circuit  is 

interrupted,  a  secondary  cur- 
rent traverses  the  secondary 

circuit  of  which  the  person  forms  a  part.     The  resulting  shocks 

will  be  quite  appreciable,  though  the  primary  current  be  produced 

by  only  a  single  small  cell. 

As  in  the  first  experiment,  the  effect  on  the  person  will  become 

more  marked  as  the  interruptions  increase  in  frequency. 

In  the  third  experiment,  the  magnetic  currents  of  the  iron  core 

add  their  inductive  influence,  as  already  explained,  to  that  of  the 

primary  current,  thus  increasing  the  intensity  of  the  secondary 

currents. 

The  effect  of  the  extra  currents  also  should  not  be  overlooked. 

As  these  traverse  the 
primary  coil,  alternat- 
ing with  each  other  in 
direction,  they  materi- 
ally modify  the  effects 
of  the  primary  and  mag- 
netic currents,  which 
are  uniform  in  direc- 
tion. 

717.  Ruhmkorff's 
Coil. — The  celebrated 
Ruhmkorff  coil  (Fig. 
407)  is  not  essentially 
different  from  the  one 


454  DYNAMICAL    ELECTRICITY. 

just  described,  except  in  having  (1)  an  arrangement  for  producing 
a  continued  and  rapid  succession  of  interruptions  in  the  primary 
current,  called  a  contact  breaker ;  this  is  a  coarsely-toothed 
wheel,  shown  at  b,  the  teeth  of  which,  on  its  being  turned,  lift  a 
lever  and  so  interrupt  the  circuit.  The  wires  from  the  battery  to 
the  primary  coil  are  attached  to  the  binding  posts  as  shown,  and 
thence  the  current  flows  to  the  contact  breaker. 

(2.)  A  condenser,  consisting  of  sheets  of  tin-foil  separated  by 
oiled  silk  or  other  insulator.  The  first,  third,  fifth,  &c.,  sheets 
are  connected  together,  and  also  with  one  end  of  the  primary 
coil,  while  the  even-numbered  sheets  are  in  like  manner  joined  to 
the  other  end  of  the  same  coil.  When  the  circuit  is  broken  the 
direct  extra  current  in  the  primary  wire  spends  its  force  in  charg- 
ing the  two  sets  of  sheets,  one  set  positively  and  the  other  nega- 
tively, and  this  condenser  instantly  discharging  again  sends  a 
reverse  current  through  the  primary,  demagnetizing  the  core  of 
wires  and  preparing  the  apparatus  for  the  next  break  of  circuit. 

The  capacity  of  a  condenser  which  holds  1  weber  (or  coulomb, 
as  it  is  proposed  to  call  it)  when  charged  to  a  potential  of  1  volt 
is  called  a  farad.  This  unit  is  too  large  for  practical  use  and 
hence  the  microfarad  is  taken.  A  condenser  of  one  microfarad 
capacity  would  require  nearly  1000  sq.  inches  of  tin-foil,  or  ten 
sheets  10  inches  square. 

718.  Power  of  the  Ruhmkorff  Coil. — The  efficiency  of  a 
Ruhmkorff  coil  depends  largely  on  complete  insulation ;  and,  in 
different  coils,  varies  greatly  with  the  length  and  fineness  of  the 
secondary  wire. 

To  secure  insulation,  the  wires  are  (as  usual)  wound  with  silk 
thread,  then  each  individual  coil  around  the  axis  is  separated 
from  the  succeeding  one  by  a  layer  of  melted  shellac,  and  lastly 
a  cylinder  of  glass  is  placed  between  the  primary  and  secondary 
coils. 

With  regard  to  the  secondary  coil,  one  of  the  largest  size  con- 
tains 280  miles  of  the  finest  copper  wire.  With  such  an  apparatus 
and  with  a  primary  current  from  thirty  quart  Grove  cells  a  spark 
of  42£  inches  has  been  obtained  ;  indeed,  all  the  tension  effects  of 
a  large  electrical  machine,  as  well  as  the  quantity  effects  of  a  power- 
ful galvanic  battery,  may  be  reproduced. 

Great  care  should  be  taken  in  handling  a  large  induction  coil, 
for  the  shock  resulting  from  its  discharge  through  the  body  would 
be  dangerous,  and  might  possibly  prove  fatal. 

719.  One  Coil  moved  into,  and  out  of,  another. — In  all 

that  has  preceded,  the  interruptions  of  the  primary  current  have 
been  supposed  to  take  place  instantaneously.  If  these  interrup- 


CHANGES    IN     THE    PRIMARY    CURRENT.       455 


FIG.  408. 


tions  are  gradual,  the  resulting  induced  currents  remain  the  same 
in  direction  as  before,  but  vary  in  intensity  and  duration. 

Thus,  if  the  primary  coil  c  (Fig.  408)  be  made  to  fit  loosely  in 
the  secondary  coil  d,  and 
then  be  moved  up  and 
down  (the  primary  circuit 
remaining  closed),  it  will 
be  found — 

(1.)  That  each  insertion 
and  removal  of  it  corres- 
ponds, the  one  to  a  gradual 
closing,  the  other  to  a 
gradual  opening  of  the 
primary  circuit—  the  result 
of  the  former  being  an  in- 
verse secondary  current,  of 
the  latter  a  direct  secondary 
current. 

And  since  a  continuous 
motion  of  the  primary  coil 
produces  a  continuous  series  of  instantaneous  secondary  currents 
with  no  appreciable  interval  between  them,  it  will  be  found — 

(2.)  That  the  secondary  currents  are  continuous  in  effect  as 
long  as  the  motion  of  the  primary  coil  is  continuous ; 

(3.)  That  their  intensity  varies  with  the  rate  of  motion  of  the 
primary  coil,  diminishing  or  increasing  as  that  is  moved  slowly 
or  rapidly  ;  from  which  it  follows — 

(4.)  That  they  cease  whenever  the  primary  coil  is  brought  to  a 
state  of  rest  in  any  position. 

720.  Changes  of  Intensity  in  the  Primary  Current.— 

All  the  results  just  mentioned  may  be  obtained  if,  instead  of 
changing  the  position  of  the  primary  coil,  as  above,  it  remain  at 
rest  while  a  corresponding  series  of  variations  be  produced  in  the 
primary  current, — an  increase  of  intensity  in  that  corresponding 
to  an  insertion  of  the  coil,  and  a  decrease  to  a  removal  of  it. 

If  two  flat  spirals,  one  of  coarse  wire  and  one  of  fine,  carefully 
insulated,  be  placed  opposite  each  other  and  near  together,  and 
the  ends  of  the  coarse  or  primary  wire  be  connected  with  the 
coatings  of  a  Leyden  jar,  a  current  will  be  induced  in  the  secondary 
spiral  at  the  instant  of  discharge,  thus  adding  another  to  the  many 
evidences  of  the  identity  of  frictional  and  dynamical  electricity. 

721.  Lenz's  Law. — If  two  conductors,  A  and  JS,  in  one  of 
which,  A,  a  current  is  flowing,  be  made  to  change  their  relative 
positions,  then  a  current  will  be  induced  in  B  in  a  direction  which 


456  DYNAMICAL    ELECTRICITY. 

will  cause  a  mutual  action  in  the  two  conductors  tending  to  oppose 
their  motion.  Thus,  if  A  and  B  be  brought  nearer  together  an 
inverse  current  will  flow  in  B,  and  currents  flowing  in  opposite 
directions  repel  each  other  (Art.  695)  ;  and  if  A  and  B  be  caused 
to  move  apart,  then  a  direct  secondary  current  will  flow  in  B,  and, 
according  to  Art.  695,  currents  flowing  in  the  same  directions 
attract  each  other.  This  statement  of  the  results  of  experiments 
will  aid  the  memory  in  regard  to  the  directions  of  the  primary 
or  secondary  currents. 

II.  CURRENTS  INDUCED  BY  MAGNETS. 

722.  Magneto-Electricity. — Faraday  reasoned  that  if  cur- 
rents could  induce  magnetism,  a  magnet  ought  to  induce  currents. 
This  he  found  to  be  the  case,  and  thus  discovered  a  new  branch 
of  physical  science,  to  which  he  gave  the  name  of  magneto-elec- 
tricity. 

If  a  magnet  be  used  instead  of  the  primary  coil  in  Fig.  408, 
all  the  phenomena  mentioned  in  Art.  719  may  be  reproduced. 

Thus,  with  the  coil  and  magnet  in 
Fig.  409,  we  obtain  the  following  re- 
sults : 

(1.)  When  the  magnet  is  alternately 
inserted  in  and  withdrawn  from  the 
coil,  the  latter  is  traversed  by  induced 
currents  alternating  with  each  other 
in  direction. 

(2.)  These  currents  are  continuous 
while  the  magnet  is  in  motion.  • 

(3.)  Their  intensity  diminishes   or 
increases  as  the  magnet  moves  slowly  or 
rapidly. 
(4.)  They  cease  when  the  motion  of  the  magnet  ceases. 

723.  Explanation  of  the  Foregoing  Phenomena.— On 

the  basis  of  Ampere's  theory,  the  correspondence  of  these  phe- 
nomena with  those  of  Art.  719  can  readily  be  accounted  for.  For 
the  magnet  may  be  considered  a  true  primary  coil,  its  magnetic 
currents  corresponding  to  the  primary  current  in  Fig.  408.  With 
regard  to  them,  the  induced  currents  are  regular  inverse  and 
direct  secondaries ;  for  in  any  given  case  they  will  be  found  to 
have  the  same  direction  as  those  induced  by  a  primary  current 
whose  direction  corresponds  with  the  supposed  direction  of  the 
magnetic  currents. 

It  will  be  seen  at  once  what  a  strong  argument  is  here  fur- 
nished in  favor  of  Ampere's  theory. 


ARAGO'S    ROTATIONS.  457 

724.  An  Iron  Core  Changing  its  Magnetic  Intensity. — 

Replace  the  magnet  (Fig.  409)  by  a  bar  of  soft  iron  inserted  in  the 
coil,  and  let  a  magnet  be  alternately  brought  near  this,  and 
removed  from  it,  as  in  Fig.  410.  The  same  results  will  be  obtained 
as  in  the  preceding  series 
of  experiments.  The  prox- 
imity of  the  magnet  induces 
magnetism  in  the  soft  iron, 
and  its  motions  to  and  fro 
produce  variations  in  this 
induced  magnetism  corres- 
ponding precisely  with  the 
varying  intensity  of  the 
primary  current  mentioned 
in  Art.  720,  and,  as  might 
be  expected,  the  results  are 
the  same. 

In  this  experiment,  it 
is  obviously  immaterial 
whether  the  coil  be  at  rest  and  the  magnet  be  moved,  or  the 
magnet  be  at  rest  and  the  coil  be  moved.  The  latter  method  is 
adopted  in  some  magneto-electric  machines. 

If  the  core  had  been  magnetic  and  a  soft  iron  disk  had  been 
moved  toward  and  away  from  its  pole,  the  mutual  induction 
would  have  varied  the  strength  of  the  magnet  and  would  thus 
have  produced  currents  in  the  coil ;  if  these  currents  had  been  led 
away  to  a  second  coil  surrounding  a  magnet,  near  whose  pole 
a  second  disk  was  held,  the  variations  of  the  strength  of  this 
magnet  would  produce  in  this  second  disk  motions  corresponding 
to  those  of  the  first.  Thus  in  the  telephone  (Art.  338),  the  vibra- 
tions of  the  transmitting  disk  are  reproduced  in  the  disk  of  the 
receiving  instrument. 

725.  Arago's   Rotations. — In   1824  Arago  observed  that 
the  oscillations  of  a  magnetic  needle  were  reduced  in  number  by 
suspending  a  copper  plate  above  it.    This  observed  phenomenon 
soon  led  him  to  the  discovery  that  if  a  horizontal  copper  disk  be 
made  to  rotate  rapidly,  a  magnetic  needle  suspended  above  it 
would  rotate  also.     This  effect  may  also  be  produced  with  other 
metals  though  in  less  degree. 

If  a  disk  of  copper  be  set  spinning  on  an  axis,  between  the 
poles  of  a  powerful  electro-magnet  whose  circuit  is  broken,  the 
axis  of  the  disk  being  parallel  to  the  lines  of  force,  the  rotation 
continues  with  slight  loss  of  velocity  for  a  long  time  ;  but  if  the 
circuit  be  suddenly  closed  the  rotation  is  at  once  checked,  or 


458 


DYNAMICAL    ELECTRICITY. 


possibly  stopped.  If  such  a  disk  be  kept  in  rapid  rotation  by  a 
suitable  band  and  pulley,  after  the  circuit  is  closed,  the  disk  will 
be  heated  by  the  action  of  the  magnet. 

These  effects  were  explained  by  Faraday  as  being  due  to  cur- 
rents induced  in  the  mass  of  metal.     Thus  let  a  needle  N  S  (Fig. 
411)  be  suspended  above  a  metal  disk  A  B.     The  magnetic  cur- 
rents flow  around  the  needle  as  indicated  in 
the  figure,  the   currents  below  the    needle  IG'  411* 

from  right  to  left  as  shown  by  the  dotted 
arrow,  and  those  above  from  left  to  right,  as 
shown  by  the  full  arrow  (Art.  700).  Now 
suppose  the  disk  to  be  rotated  in  the  direc- 
tion from  A  to  B ;  the  portions  of  the  cur- 
rents around  N  8  which  are  nearest  to  the 
disk  will  induce  in  that  part  of  the  disk 
towards  A  currents  whose  directions  are 
such  as  to  resist  the  motion  of  the  disk,  according  to  Lenz's  law 
(Art.  721),  that  is  to  say,  currents  will  flow  in  the  disk  from  left 
to  right ;  while  in  that  part  of  the  disk  towards  B,  which  is  mov- 
ing away  from  JV,  the  induced  currents  are  from  right  to  left, 
and  so  resist  the  motion  of  B  aivay  from  N. 

If  the  needle  had  been  moved,  the 

FIG.  412.  disk  remaining  fixed,  the  same  analysis 

of  the  motion  might  be  made,  and 
we  should  find  that  the  disk  would  re- 
sist the  motion  of  the  needle.  A 
copper  collar  or  frame  is  sometimes 
used  to  coil  the  galvanometer  wire 
upon,  in  order  to  reduce  or  damp  the 
oscillations  of  the  needle,  and  bring  it 
more  quickly  to  rest. 

726.  Clarke's  Magneto-electric 
Machine.  —  In 

front  of  the  poles 
ofthell-magnet, 
A  (Fig.  412),  is 
revolvedthearw- 
ature,  consisting 
of  the  two  bob- 
binSy  B9  B' ,  which 
are  coils  of  fine 
wire  with  cores 
of  soft  iron. 
These  cores  are  joined  to  each  other  and  to  the  axis  of  r  station 


THE    COMMUTATOR. 


459 


FIG.  413. 


by  the  bar  of  soft  iron,  V,  and  motion  is  communicated  by  the 
multiplying  wheel  and  band  at  W. 

As  one  of  the  bobbins  passes  before  a  north  pole  while  the 
other  is  passing  before  a  south  pole,  the  resulting  induced  cur- 
rents are  relatively  of  contrary  directions;  but  as  one  of  the 
coils  is  always  right-handed  and  the  other  left-handed,  the  cur- 
rents passing  through  them  at  any  given  instant  have  the  same 
absolute  direction,  so  that  the  two  coils  act  as  one. 

Fig.  413  shows  the  poles  of  the  fixed  magnet,  and  the  direction 
in  which  the  armature  revolves.  The  maximum  magnetization 
of  the  soft  iron  cores  occurs  when  the  bobbins  are  directly  in  front 
of  JVand  S.  While  they  move  through 
the  first  and  third  quadrants  they  are 
losing  their  magnetism,  and  while  mov- 
ing through  the  second  and  fourth  they 
are  acquiring  that  of  the  contrary  kind. 
The  resulting  induced  currents  will  thus 
be  direct  and  inverse  to  contrary  kinds 
of  magnetism,  and  will  therefore  have 
the  same  absolute  direction.  But  as  the 
bobbins  pass  from  the  second  quadrant 
to  the  third,  and  from  the  fourth  to  the 
first,  they  lose  the  magnetism  just  ac- 
quired, and  the  induced  currents  change 
from  inverse  to  direct  with  reference  to 
the  same  kind  of  magnetism,  and  there- 
fore become  reversed  in  absolute  direc- 
tion. 

Hence  the  semi-revolutions  of  the 
armature  on   opposite  sides  of  a  line 

joining  the  poles  of  the  permanent  magnet  produce  currents  of 
contrary  directions. 

FIG.  414. 


art 


727.  The  Commutator. — Fig.  414  is  an  enlarged  view  of 
the  outer  end  of  the  axis  (Fig.  412),  and  shows  the  commutator,  or 


460  DYNAMICAL    ELECTRICITY. 

arrangement  by  means  of  which  the  contrary  currents  just  men- 
tioned are  made  to  furnish  one — or  rather  a  series  of  currents — 
flowing  in  the  same  direction. 

Two  pieces  of  brass,  m  and  n,  fastened  to  the  axis  and  revolv- 
ing with  it,  are  insulated  from  each  other  by  being  fastened  to  an 
ivory  ring,  i,  around  the  axis,  and  are  connected  with  the  ends  of 
the  wires  of  the  coils  ;  that  is,  they  are  made  the  poles  of  the  two 
coils  acting  as  one.  Against  m  and  n  press  two  springs,  b  and  c, 
which  are  attached  to  the  plates  P  and  P',  insulated  from  each 
other,  from  which  plates  the  wires  are  led  to  the  handles  h  h'.  If 
the  handles  h  and  h'  be  joined,  the  circuit  will  be  complete  when 
m  and  n  in  their  revolutions  press  against  b  and  c.  Let  us  sup- 
pose that  the  induced  current  is  passing  in  the  direction  indicated 
by  the  arrows.  When  the  armature  has  revolved  through  180° 
from  its  present  position,  m  and  n  will  have  changed  places — but 
they  tuill  also  have  changed  polarities  (Art.  726).  Therefore  n 
presents  to  b  the  same  polarity  which  m  did,  and  hence  there  is 
no  change  in  the  direction  of  the  current  through  b  and  c. 

728.  Effects  of  Rapid  Revolution.— The  intensity  of  the 
induced  currents  of  this  machine,  as  also  the  rapidity  with  which 
they  succeed  each  other,  is  regulated  by  the  rate  of  revolution  of 
the  armature.    When  this  is  rapidly  revolved,  they  produce  all  the 
effects  of  a  single  voltaic  current,  so  that  the  apparatus  may  be 
used  as  a  galvanic  battery  with  h  and  h'  for  its  electrodes.     At  the 
same  time  its  physiological  effects  are  most  remarkable,  the  shocks 
becoming  unendurable  when  it  is  revolved  with  great  rapidity. 

729.  Large  Machines. — In  large  magneto-electric  machines 
of  this  kind,  increased  efficiency  is  obtained  in  two  ways  : 

First,  by  multiplication  of  magnets.  In  Nollet's  machine,  con- 
structed in  1850,  192  magnetized  steel  plates  are  so  combined  as 
to  make  40  powerful  U-magnets.  These  are  arranged  in  eight 
rows  around  the  circumference  of  a  large  iron  frame  inside  of 
which  revolve  sixty-four  bobbins. 

Second,  by  multiplication  of  currents.  In  Wild's  machine,  con- 
structed on  this  plan,  the  induced  currents  first  obtained,  instead 
of  being  directly  utilized,  are  passed  through  the  coils  of  a  large 
electro-magnet.  Before  the  poles  of  this  a  second  armature  re- 
volves, and  the  resulting  induced  currents  are  far  more  powerful 
than  the  first.  These  may  in  turn  be  made  to  magnetize  a  second 
electro-magnet  before  the  poles  of  which  a  third  armature  re- 
volves, &c. 

Currents  of  very  great  intensity  may  be  obtained  from  either 
of  these  machines.  The  motive  power  employed  is  generally  a 
steam-engine  of  from  one  to  fifteen-horse  power. 


CUTTING    LINES    OF    FORCE.  401 

730.  Motion  of  a  Straight  Conductor  through  Lines 
of  Force.  —  A  magnetic  field  has  been  defined  (Art.  577)  and  the 
direction  of  the  lines  of  force  has  been  studied.  If  now  a  con- 
ducting wire,  perpendicular  to  the  lines  of  force,  be  moved  in  a 
direction  perpendicular  to  its  own  length,  so  as  to  cut  new  lines 
of  force  at  each  instant,  a  current  will  be  generated  in  the  moving 
wire,  provided  the  circuit  be  closed  outside  the  magnetic  field  ;  if 
the  circuit  be  not  complete,  then  a  difference  of  potential  at  the 
two  ends  of  the  wire  will  result.  If  the  magnetic  field  be  moved 
while  the  wire  remains  at  rest,  the  same  effects  will  follow.  If  the 
wire  and  the  field  be  both  at  rest,  but  if  the  strength  of  the  field 
be  increased,  a  current  will  flow,  since  the  increase  of  the  magnetic 
field  adds  new  lines  of  force  which,  crowding  together  as  it  were, 
are  cut  by  the  wire  ;  a  decrease  of  the  strength  of  the  field  would 
cause  a  loss  of  lines  of  force,  and  a  separating  of  those  left,  and 
so  a  movement  of  the  lines  across  the  wire,  resulting  in  a  reverse 
current.  The  strength  of  the  current  generated  depends  on  the 
number  of  lines  of  force  cut  in  a  unit  of  time  ;  hence,  for  a  given 
field  it  varies  with  the  velocity  of  motion  of  the  conductor,  and 
increases  as  the  length  of  the  part  of  the  conductor  acted  upon  by 
the  field  increases  ;  for  a  given  velocity  and  length  of  conductor, 
it  varies  with  the  strength,  of  the  field  ;  that  is,  it  varies  in  pro- 
portion to  the  closeness  of  the  lines  of  force. 

Let  E  represent  the  electromotive  force,  H  the  strength  of  the 
magnetic  field,  I  the  length  of  the  conductor  acted  upon,  L  the 
distance  moved,  and  T  the  time,  and  we  have 


But  -m  equals  the  velocity  of  the  conductor,  therefore  the  electro- 

motive force,  per  unit  of  length  of  conductor,  is  equal  to  the  strength 
of  the  field  multiplied  by  the  velocity  of  motion  of  the  conductor. 

If  the  conductor  be  moved  in  the  direction  of  the  lines  of  force 
no  current  will  result. 

By  applying  Lenz's  law  the  direction  of  the  induced  current 
may  be  determined.  Let  the  student  suppose  himself  to  represent 
any  line  of  force,  his  feet  being  upon  the  marked  pole  of  a  mag- 
net ;  then,  holding  a  wire  conductor  in  his  outstretched  hands, 
upon  moving  the  wire  forward  through  the  magnetic  field,  in  the 
direction  in  which  he  faces,  a  current  will  flow  from  his  left 
hand  to  his  right.  A  wire  moved  through  the  earth's  magnetic 
field  at  right  angles  to  the  direction  of  the  dipping  needle  will 
deflect  the  galvanometer  needle. 

731.  Motion  of  a  Coiled   Conductor.—  If  instead  of  a 


462  DYNAMICAL    ELECTRICITY. 

straight  conductor  a  coil  of  wire  be  used  the  effect  is  as  follows : 
Suppose  a  coil  to  be  moved  towards  the  north  pole  of  a  magnet, 
as  in  Fig.  415,  the  lines  of  force  being  represented  by  dotted  lines. 

As  the  coil    approaches    the 

FIG.  415.  p0ie  jt  js  seen  that  new  lines 

of  force  are  cut  at  each  in- 
stant, a  greater  number  thus 
being  included  within  it ;  and 
regarding  any  limited  portion 
of  the  coil  as  straight,  it  fol- 
lows from  what  precedes  that 
a  current  will  flow  as  indicated  by  the  arrows.  After  passing  the 
middle  of  the  magnet  negative  lines  of  force  will  be  cut,  and  the 
current  will  be  reversed,  as  shown.  If  the  motion  of  the  coil  had 
been  away  from  the  north  pole  in  the  first  instance  fewer  and 
fewer  positive  lines  of  force  would  have  been  included  within  it, 
and  a  reverse  current  would  have  resulted. 

732.  Dynamo-Electric  Machines.— ^These  machines  differ 
from  the  magneto-electric  machines  previously  described,  in  that 
the  whole,  or  a  part,  of  the  current  from  the  revolving  ring  arma- 
ture is  made  to  pass  through  the  coils  of  the  field  electro-magnets, 
the  slight  residual  magnetism  of  which  is  sufficient  to  start  a 
feeble  current  in  the  armature,  which  current  instantly  strengthens 
the  magnetic  field,  and  the  machine  is  thus  soon  worked  up  to  its 
full  efficiency. 

The  Gramme  Machine  may  be  taken  as  a  type,  and  the  expla- 
nation of  its  working  will  lead  to  the  understanding  of  other 
applications  of  the  principle  involved.  Let  two  half-ring  magnets 
be  placed  with  their  like  poles  in  con- 
tact, as  in  Fig.  416,  and  suppose  the 
coil  G  to  be  moved  from  the  neutral 
point  A  towards  the  poles  at  JV.  From 
the  last  article  it  will  be  seen  that  a  cur- 
rent will  be  induced  in  the  direction  of 
the  arrow,  and  during  the  progress  of 
the  coil  from  A  through  N  to  B  the 
current  will  not  change  direction,  but 
will  increase  in  strength  by  being  sub- 
ject to  the  same  inductive  influence 
during  the  whole  progress  of  the  coil.  On  passing  the  neutral 
point  B,  the  current  will  be  reversed,  as  shown  by  the  arrows,  and 
will  flow  in  the  inverse  direction  during  the  movement  of  the  coil 
from  B  through  S  to  A.  Now,  as  it  is  not  convenient  to  move 
coils  around  a  ring,  let  us  move  the  poles  JVand  S  around  within 


DYNAMO-ELECTRIC    MACHINES. 


463 


FIG.  417. 


a  set  of  coils  placed  on  an  iron  ring  at  the  points  1,  2,  3,  &c. 

This  we  may  accomplish  by  causing  the  iron  ring,  with  its  coils, 

to  rotate  between  the  poles  of  a  magnet,  in  the  direction  of  the 

arrow  (Fig.  417).     The  in- 

stanta neons  pole  JVremaining 

always  opposite  the  pole  S' 

of    the    inducing    magnet, 

while  2  and  1  are  made  to 

approach  it  in  succession  by 

the   rotation    of    the    ring, 

causes      currents    precisely 

similar  to  those  obtained  by 

moving  the  coil. 

Now  let  us  examine  the 
effect  of  the  stationary  pole  8'  upon  the  moving  coils.  It  was 
shown  in  the  last  article  that  a  coil  approaching  a  S  pole  so  as  to 
include  more  and  more  lines  of  force  would  have  a  current  flowing 
in  it  from  right  over  to  left,  as  one  faces  the  direction  of  motion, 
or  contrary  to  the  motion  of  the  hands  of  a  watch,  and  if  it  were 
moved  away  from  the  pole  so  as  to  inclose  fewer  lines  of  force  the 
current  would  flow  from  left  over  to  right.  Now  as  the  coil 
moves  from  1  to  2  its  plane  becomes  more  nearly  parallel  to  the 
the  lines  of  force,  and  therefore  the  number  of  these  inclosed 
decreases,  until  at  S'  its  plane  coincides  with  their  direction  and 
none  are  inclosed.  The  current  due  to  S'  is  therefore  in  the 
direction  of  the  arrows  as  before.  After  passing  S'  the  coil 
incloses  more  and  more  lines  of  force,  and  therefore  the  direction 
of  the  current  should  be  from  left  over  to  right  as  in  the  figure. 
Thus  it  is  seen  that  the  stationary  pole  S'  and  the  constantly- 
shifting  pole  N  botli  con- 
spire to  produce  currents 
of  the  same  kind  in  the 
coils  from  A  to  B ;  and 
also  that  N'  and  S  pro- 
duce reverse  currents  in 
the  coils  from  B  to  A. 

Now  suppose  a  con- 
tinuous wire  to  be  wound 
around  the  ring  armature, 
as  in  Fig.  418.  It  is  evi- 
dent, if  the  student  has 
followed  the  explanation  thus  far,  that  a  current  flows  from  A 
through  N  to  B,  and  from  A  through  S  to  B',  hence  if  we  place 
stationary  metal  strips  m  and  n  not  attached  to  the  revolving  ring, 
so  that  each  may  be  in  contact  with  two  of  the  short  wires  soldered 


FIG.  418. 


464 


DYNAMICAL    ELECTRICITY 


to  the  inner  points  of  the  spiral  at  any  given  moment,  a  continu- 
ous current  will  flow  through  a  wire  of  which  these  metal  strips 
are  the  terminals,  and  this  current  may  be  carried  around  the 
field  magnets  before  being  led  off  for  use,  as  shown  in  the  figure. 
In  the  Gramme  machine  the  wire  is  coiled  in  sections,  and  these 
are  united  into  one  continuous  coil  by  attaching  the  end  of  one 
coil,  and  the  beginning  of  the  succeeding  one,  to  a  copper  slip 
fastened  upon  a  cylinder  of  insulating  material  and  parallel  to  its 
axis.  These  copper  terminals,  in  number  equal  to  the  number  of 
coiled  sections,  are  insulated  from  each  other,  and  perform  the 
office  of  the  short  wires  used  for  illustration  in  Fig.  418.  Two 
slips  of  copper  press  upon  the  revolving  cylinder  of  terminals,  one 
above  and  one  below,  and  lead  off  the  current. 

Fig.  419  represents  one  form  of  Gramme  machine.     When 

FIG.  419. 


great  electromotive  force  is  required,  as  in  arc  electric  lighting,  it 
is  obtained  by  increasing  the  number  of  turns  of  wire  in  the 
rotating  coils,  and  by  increasing  the  velocity  of  rotation  within 
due  limits.  When  currents  of  feeble  electro-motive  force,  but 
of  considerable  strength  are  required,  as  in  electro-plating,  the 
internal  resistance  must  be  small,  and  hence  the  revolving  coil 
must  be  a  few  turns  of  stout  wire. 

The  continuous  current  dynamo-machines  become  electro- 
motors when  driven  by  a  current  from  a  battery,  or  from  another 
dynamo. 


ELECTROLYSIS.  465 


CHAPTER    IV. 

PRACTICAL    APPLICATIONS. 

733.  Classification.— The  applications  of  Galvanic  electri- 
city in  the  arts  and  sciences,  as  well  as  in  the  affairs  of  every  day 
life,  are  eminently  practical.     They  may  be  classified  according 
to  the  way  in  which  are  utilized  those  molecular  forces  whose 
resultant  is  known  as  the  current. 

It  may  be  stated  in  general  that  these  applications  are  made 
either  within  the  circuit,  or  exterior  to  it. 

Within  the  circuit.  Here  the  electrical  force  is  employed  (I) 
directly,  as  in  chemical  decomposition,  or  (II)  by  being  first  made 
to  produce  the  effects  light  and  heat,  which  are  then  applied  as 
desired. 

Without  the  circuit.  Here  it  is  employed  indirectly  (III)  by 
being  made  to  reappear  as  mechanical  motion  through  the  inter- 
vention of  the  kindred  force,  magnetism. 

Examples  will  be  given  of  each. 

I. — DIRECT  APPLICATIONS   OF  THE  CURRENT. 

734.  Electrolysis. — When  a  current  is  passed  through  a 
binary  compound  (i.  e.,  one  containing  two  elements),  the  com- 
pound is  decomposed,  one  of  its  elements  appearing  at  the  positive 
electrode,  the  other  at  the  negative. 

For  instance,  water,  con-  FIG.  420. 

sisting  of  the  two  gases 
oxygen  and  hydrogen,  is 
thus  decomposed. 

In   the  bottom  of    the 
dish  D  (Fig.  420),   partly 
filled  with  water,  are  fast- 
ened p  and  n,  the  platinum 
electrodes     of    a    battery. 
Over  these  are  placed  two 
tubes,    0   and  H,  full    of 
water.     On  closing  the  circuit,  oxygen  rises  from  p  into  0,  and 
hydrogen  from  n  into  H. 
30 


466  DYNAMICAL     ELECTRICITY. 

Electrolysis  is  of  the  utmost  importance  in  chemistry.  Thus, 
the  preceding  experiment  gives  a  correct  analysis  of  water,  and  if 
oxygen  had  been  previously  unknown,  would  have  been  the  means 
of  its  discovery.  In  this  way  were  discovered  several  of  the 
metallic  elements. 

735.  Voltameter. — The  quantity  of  any  compound  decom- 
posed in  a  given  time  is  proportional  to  the. strength  of  the  current ; 
and  hence  by  measuring  the  quantity  of  water  decomposed  in  a 
given  time,  and  reducing  the  resulting  volume  of  mixed  gases  to 
a  standard  temperature  and  pressure,  the  strength  of  the  current 
may  be  determined.    An  instrument  for  this  purpose  is  called  a 
Voltameter. 

The  objections  to  its  uso  are  that  it  gives  the  total  of  current 
during  the  time,  but  not  the  strength  at  any  given  instant,  and 
hence  the  current  might  be  very  variable  during  the  time  of  de- 
composition, and  yet  the  instrument  would  give  no  indication  of 
this  fact ;  it  cannot  measure  currents  too  weak  to  effect  decom- 
position ;  and  its  results  are  affected  by  the  acidity  of  the  water, 
the  size  of  the  electrodes,  and  the  distance  between  them. 

736.  Electro-plating. — Electrolysis  is  very  important  in  its 
applications  in  the  arts. 

When  a  solution  of  a  metallic  salt  is  subjected  to  the  action  of 
the  current,  it  is  decomposed  and  a  permanent  film  of  the  metal 
is  deposited  on  any  suitable  material  placed  so  as  to  receive  it. 
The  process  is  then  called  electro-metallurgy,  or  electro-plating. 

The  bath  (Fig.  421)  contains  a  saturated  solution  of  blue 
vitriol  (sulphate  of  copper).  In  this  is  suspended  by  wires  from 

the  metallic  rod  D  a  plate 

FlG-  421  of  copper  C9  and  from  B 

(also  metallic)  the  cast  of 
a  medal  m,  which  is  to  be 
coated  with  copper.  Con- 
nect D  with  the  positive 
electrode  of  a  battery  and 
B  with  the  negative.  The 
current  passing  through 
the  solution  removes  from 
it  particles  of  copper,  and 
deposits  thorn  on  m.  Those  taken  from  the  liquid  are  replaced 
by  others  taken  from  0,  which  is  thus  gradually  wasted  away,  and 
the  solution  is  kept  saturated. 

If  the  bath  contains  a  solution  of  gold,  and  (7  is  replaced  by  a 
piece  of  gold  and  m  by  a  silver  cup,  the  cup  will  be  electro-gilded. 
Electro-silvering  is  an  analogous  process. 


STORAGE    CELLS.  467 

To  produce  in  any  case  a  firm  and  even  coating,  the  process 
must  be  allowed  to  proceed  slowly  by  the  employment  of  a  weak 
current.  On  a  small  scale  a  single  cell  is  sufficient.  In  large 
establishments  a  magneto-electric  machine  turned  by  steam  has 
been  successfully  and  economically  used. 

737.  Electrotyping. — By   taking  proper  precautions,  the 
copper  film  deposited  on  m  may  be  removed,  and  its  surface  will 
be  found  to  present  an  exact  fac-simile  of  the  medal  of  which  m  is 
an  impression. 

Therefore  if  m  is  an  impression,  in  wax  or  paper  pulp,  of  the 
type  from  which  a  page  is  printed,  the  mould  or  impression  hav- 
ing been  coated  with  fine  plumbago  to  render  it  a  good  conductor, 
the  deposited  copper  may  be  removed,  and  having  been  stiffened 
by  melted  lead  (or  some  alloy)  poured  over  its  under  surface,  it 
may  be  used  in  the  printing-press  instead  of  the  type.  It  is  then 
called  an  electro-type  plate,  and  when  not  in  use  may  be  preserved 
indefinitely  for  succeeding  editions,  while  the  type  of  which  it  is  a 
copy  can  be  distributed  and  used  for  other  purposes. 

738.  Secondary  Battery. — In  Art.  C86  it  was  stated  that 
single  fluid  cells  become  polarized  by  the  accumulated  hydrogen, 
and  a  reverse  current  is  set  up.     If  the  platinum  electrodes  of  a 
voltameter  be  disconnected  from  the  battery,  and  connected  with 
i  galvanometer,  the  deflection  of  the  needle  indicates  the  flow  of 
a  reverse  current,  due  to  the  condensation  of  oxygen  upon  the 
positive  electrode  and  hydrogen  upon  the  negative. 

Planters  Secondary  Cell  consists  of  two  sheets  of  lead,  between 
which  is  laid  a  perforated  sheet  of  non-conducting  substance,  the 
whole  rolled  up  loosely,  and  placed  in  a  jar  containing  dilute 
sulphuric  acid  as  in  Fig.  422.  The  strip  of  copper,  a,  is  connected 
with  one  of  the  lead  sheets  and  b  with  the  other.  If  the  poles  of  a 
battery  of  several  Grove  cells  in  series  be  connected 
with  a  and  I,  the  sheet  connected  with  the  positive 
pole  becomes  coated  with  peroxide  of  lead  while 
the  surface  of  the  other  is  deoxidized  or  reduced  to 
the  metallic  state.  In  this  polarized  condition  the 
cell  is  capable  of  giving  a  reverse  current,  which 
will  flow  until  the  surfaces  of  the  lead  sheets  are 
reduced  to  identical  chemical  condition  again. 
The  electromotive  force  of  such  cells  is  about  2.5 
volts,  and  the  cells  will  retain  their  charge  many 
days  with  but  slight  loss.  As  the  Plante  cell 
requires  many  chargings  and  reversals  to  reduce  the 
lead  surfaces  to  a  spongy  condition  and  produce  its  maximum 
efficiency,  a  modified  form  devised  by  Faure  is  more  generally 


468 


DYNAMICAL    ELECTRICITY. 


used.  In  this  cell  the  lead  plates  of  the  accumulator  are  coated 
with  red  lead  at  the  outset,  and  cells  thus  prepared  need  but  little 
"forming,"  as  the  process  of  charging  and  reversals  is  called. 

739o  Medicinal  Applications. — The  shocks  produced  by 
the  passage  of  interrupted  currents  through  the  system  have 
already  been  alluded  to.  In  certain  ailments  these  shocks,  when 
properly  applied,  have  been  known  to  produce  beneficial  results. 
On  the  other  hand,  great  injury  has  resulted  from  their  misappli- 
cation. Hence  they  should  be  employed  as  remedies  only  under 
the  direction  of  a  reliable  physician. 

The  familiar  medical  magneto-electrical  machine,  which  comes 
compactly  stored  in  a  box  ten  inches  long  and  about  four  inches 
square  at  the  end,  does  not  differ  essentially  from  Clarke's  (Art. 
726),  except  in  lacking  the  commutator,  so  that  its  currents  pass 
through  the  body  alternating  with  each  other  in  direction. 


II.  APPLICATIONS  OF  ELECTRIC  LIGHT  AND  HEAT. 

740.  Light  by  the  Electric  Current.— The  electric  light 
may  be  advantageously  employed  for  brilliant  illumination ;  also 
where  a  strong  penetrating  light  is  needed,  as  in  light-houses,  or 
for  signals  between  ships.  For  exhibiting  to  an  audience  magni- 
fied images  of  small  objects  (as  with  the  projecting  microscope)  it 
has  no  superior  ;  and  to  the  physical  experimenter  the  various 
colors  it  assumes  on  passing  through  highly  rarefied  gases  of  dif- 
pIG  423.  ferent  kinds  are  of 

great  interest. 

To  obtain  the  most 
brilliant  effects  car- 
bon electrodes  must 
be  employed,  and  as 
these  are  constantly 
changing  in  length 
they  must  be  kept  at 
a  uniform  distance 
apart  by  machinery. 
The  flame  is  not 
straight,  but  curved, 
as  in  Fig.  423,  and 
is  called  the  voltaic 
arc.  To  obtain  it, 
the  electrodes  must 
first  be  made  to 


ELECTRIC    LIGHTING.  469 

touch  each  other.  With  92  Bunsen  elements  the  light  has  been 
found  to  possess  more  than  one-third  the  intensity  of  direct  sun- 
light. 

In  the  modern  applications  of  electric  lighting,  two  systems 
are  advocated  ;  in  one  the  voltaic  arc  is  employed  and  in  the 
other  an  incandescent  conductor.  In  the  Jabloclikoff  candle  two 
rods  of  carbon  placed  parallel  are  separated  by  a  layer  of  a  fusible 
non-conductor,  usually  Kaolin,  which  melts  at  the  point  where 
the  voltaic  arc  is  maintained  ;  by  occasionally  reversing  the  cur- 
rent the  waste  of  the  carbons  is  equalized. 

An  incandescent  lamp  consists  of  a  filament  of  carbon  inclosed 
in  a  glass  vacuum  bulb.  When  the  current  traverses  the  filament 
it  glows  with  great  brilliancy.  The  current  for  lights  of  either 
system  is  now  furnished  by  large  dynamo  machines,  driven  by 
steam-power,  many  lamps  being  included  in  the  same  circuit. 

741.  Heat  by  the  Electric  Current.— The  heat  as  well  as 
the  light  of  the  voltaic  arc  is  intense.  In  the  laboratory  it  is 
employed  to  deflagrate  and  volatilize  refractory  substances.  When 
the  lower  electrode  is  hollowed  out  in  the  form  of  a  cup,  a  piece 
of  platinum  (one  of  the  least  fusible  of  metals)  placed  in  it  is 
melted  like  wax  in  a  candle,  and  a  diamond,  the  hardest  of  known 
substances,  is  burnt  to  a  black  cinder.  To  produce  either  of 
these  results,  a  battery  of  great  power  must  be  used. 

Metals  may  also  be  deflagrated  by  being  made  part  of  the 
circuit  in  the  form  of  very  fine  wires.  They  are  thus  employed 
to  spring  mines  in  time  of  war,  or  in  blasting  rocks.  In  Fig. 
424,  B  is  a  box  full  of  fulminating  powder,  and  w  is  a  very  fine 

PIG.  424. 


platinum  wire,  about  -|  in.  in  length,  fastened  to  p  and  n,  which 
are  insulated  copper  wires  extending  to  a  battery  situated  at  any 
convenient  distance.  B  and  its  contents  are  the  fuse  which  is 
inserted  in  the  powder  to  be  fired.  When  a  moderately  strong 
current  passes  through  w,  it  is  heated  sufficiently  to  ignite  the 
fuse,  and  the  powder  explodes. 

III.  MECHANICAL  APPLICATIONS. 

742.  Made  through  the  Medium  of  Induced  Magnet- 
ism.— As  has  been  seen  in  preceding  experiments,   magnetism 


470 


DYNAMICAL    ELECTRICITY. 


FIG.  425. 


may  be  induced  in  a  piece  of  steel  or  iron  by  the  current  passing 
through  a  circuit  which  is  near  it.  Hence  induced  magnetism 
and  its  applications  are  results  obtained  outside  of  the  circuit. 

This  magnetism  may  be  utilized  directly.  For  compass  and 
galvanometer,  needles  are  ordinarily  made  by  placing  a  steel  needle 
in  a  helix  through  which  a  current  is  sent  (Art.  709). 

But  its  most  numerous  and  important  applications  are  in  the 
way  of  mechanical  movements.  All  these  are  modifications  of  the 
simple  rising  and  falling  of  the  armature  of  a  U-magnet,  men- 

tioned in  Art.  710.  Thus,  the 
armature  A  (Fig.  425)  is  lim- 
ited in  its  fall  by  the  metallic 
base  B,  so  that  it  is  within  the 
influence  of  M  the  next  time 
that  it  becomes  a  magnet. 
Hence,  when  the  circuit  is 
closed  and  broken  at  n  p,  the 
end  A  of  the  lever  L  rises 
and  falls.  It  is  evident  that  the  corresponding  motions  of  the 
end  ^may  be  applied  in  a  variety  of  ways.  A  few  of  these  are 
described  in  the  following  articles. 

743.  Electro-magnetic  Engine.  —  E  may  be  attached  to  a 
vertical  arm,  arid  that  to  the  crank  of  a  fly-wheel  (Fig.  426),  and 
the  interruptions  of  the  current  may  be  made  automatic  by  con- 
necting p  with  L,  and  n 
with  B.  When  A  rests  on 
B  the  circuit  is  closed  ;  M 
becomes  a  magnet,  and  A  is 
attracted  by  it  ;  but  as  soon 
as  A  rises  from  B  the  cir- 
cuit is  opened,  M  no  longer 
attracts  it,  and  it  falls  back, 
only  to  close  the  circuit 
again  and  repeat  the  same 
movements  as  before.  The 
tendency  of  this  is  to  pro- 
duce a  rotar  motion  in  the 


FIG.  420. 


.TS 


fly-wheel,  and  the  apparatus 

involves  the  principle  of  a 

single-acting  engine.     With  a  second  electro-magnet,  and  a  some- 

what different  arrangement  of  parts,  an  actual  double-acting 

engine  may  be  constructed. 

In  another  form  of  engine,  a  magnet  is  drawn  into  a  hollow 
coil,  and  then,  upon  reversal  of  the  current,  repelled  again,  thus 


ELECTRO-MAGNETIC    TELEGRAPH. 


471 


acting  like  the  piston  of  a  steam-engine.  In  still  another,  a  num- 
ber of  armatures  are  arranged  upon  a  revolving  wheel,  like  the 
floats  upon  a  water-wheel,  and  these  are  attracted  in  succession  hy 
a  series  of  electro-magnets  arranged  radially  within  the  revolving 
armatures.  This  last  form  is  practically  applied,  as  in  Edison's 
pen,  to  produce  a  very  rapid  reciprocating  motion  when  but  little 
power  is  required. 

Various  other  forms  of  this  engine  have  been  constructed  and 
exhibited  as  curiosities,  or  used  where  expense  was  not  regarded. 
But  it  cannot  compete  with  the  steam-engine  as  long  as  zinc  and 
acids  cost  so  much  more  than  coal,  unless  a  practical  method  shall 
be  devised  of  transmitting  a  current  generated  by  water-power,  or 
other  natural  force,  to  long  distances  to  be  used  in  dynamo  motors. 

744.  Electro-magnetic  Telegraph. — To  our  countryman, 
Prof.  S.  F.  B.  Morse,  is  due  the  credit  of  the  erection  of  the  first 
telegraph  line  in  the  United  States.  It  extended  from  Baltimore 
to  Washington,  and  went  into  operation  in  1844. 

Communication  in  various  ways  by  means  of  electricity  between 
places  a  few  miles  apart  was  not  unknown  in  Europe  before  that 
time,  and  several  ingenious  systems  have  appeared  since  ;  but  the 
Morse  system  has  been  very  generally  preferred  on  account  of  its 
greater  simplicity  and  efficiency,  and  it  is  now  widely  used  in  the 
United  States  and  on  the  continent  of  Europe,  where  it  is  known 
as  the  American  system.  The  principle  of  its  operation  is  as  fol- 
lows : 

Let  E  of  Fig.  425  be  furnished  with  a  style  e  (Fig.  42?)  directly 

FIG.  427. 


over  which  is  the  groove  on  the  surface  of  a  solid  brass  roller  c. 
Between  c  and  e  is  the  long  paper  ribbon  R  R.  Also  let  A  be 
placed  above  M  and  be  furnished  with  a  spring  s  to  raise  it  as  far 
as  the  screw  i  allows  when  it  is  not  attracted  by  M.  When  the 
circuit  is  closed,  A  is  attracted  and  e  rises  and  forces  the  paper 
into  the  groove,  producing  a  slight  elevation  on  its  upper  surface. 


472 


DYNAMICAL    ELECTRICITY. 


The  ribbon  is  pulled  along  at  a  uniform  rate  in  the  direction  of 
the  arrow  by  clockwork  (not  shown  in  the  figure),  so  that  when 
the  circuit  remains  closed  for  a  little  time,  a  dash  is  marked  on 
the  paper  by  e  ;  when  it  is  closed  and  instantly  opened,  the  result 
is  a  dot — or  rather  a  very  short  dash.  Spaces  are  left  between 
these  whenever  the  circuit  is  opened.  Combinations  of  these 
dots,  dashes,  and  spaces,  all  carefully  regulated  in  length,  compose 
the  letters  of  the  alphabet.  Spaces  are  also  left  between  the 
letters,  and  longer  ones  between  words. 

By  lengthening  the  circuit  wire,  it  is  evident  that  the  person 
who  sends  the  message  at  n  p,  and  the  one  who  receives  it  at  E, 
may  be  miles  apart,  and  the  transmission  will  be  almost  instanta- 
neous owing  to  the  rapid  passage  of  the  current. 

The  essential  parts  of  this  system,  or  indeed  of  any  system,  are 
a  communicator  at  n  p,  an  indicator  at  E,  and  a  wire  extending 
from  one  to  the  other. 

745.  The  Connecting  Wire. — It  was  at  first  supposed  that 
a  complete  metallic  circuit  was  necessary,  hence  a  return  wire  was 
employed.  But  this  was  rejected  when  it  was  found  that  the  earth 
could  be  used  as  a  part  of  the  circuit,  as  shown  in  Fig.  428,  in 
which  the  dotted  line  and  arrow  beneath  the  surface  are  not 

FIG.  438. 


intended  to  convey  the  idea  that  a  current  actually  flows  from 
one  earth  plate  to  the  other,  but  that  a  complete  circuit  is  formed, 
the  earth  acting  the  part  of  an  infinite  reservoir  of  electricity. 
8  and  8'  are  the  terminal  stations,  and  s  is  one  of  the  way  stations 
which  may  occur  anywhere  along  the  line.  At  every  station  both 
a  communicator,  C,  and  indicator,  /,  are  introduced  into  the  cir- 
cuit, so  that  messages  can  be  both  sent  and  received. 

746.  The  Communicator. — This  consists  of  a  lever,  I  (Fig. 
429),  and  anvil,  a,  both  of  brass,  and  insulated  from  each  other. 
The  anvil  connects  with  the  line  wire  W,  and  I  with  the  rest  of 
the  circuit  through  W,  and  W  W"  of  the  next  figure.  (See 
also  Fig.  428.)  The  end  of  I  is  depressed  by  the  finger  of  the 


THE    INDICATOR. 


473 


operator  on  the  insulating  button  b,  and  is  raised  by  the  spring 
s  when  the  pressure  is  removed.    The  former  movement  closes 


FIG.  429. 


the  circuit,  the  latter  opens  it,  and  by  a  succession  of  these  the 
message  is  sent. 

When  the  communicator  is  not  in  use,  the  brass  bar  Tc  hinged 
to  the  base  of  I  is  pressed  into  contact  with  a.  This  closes  the 
circuit  for  other  stations  on  the  line,  and  hence  k  is  called  the 
circuit  closer.  The  whole  apparatus  is  called  the  key. 

747.  The  Indicator. — This  consists  of  two  parts,  (1st)  the 
relay,  and  (2d)  either  the  register,  or  the  sounder. 

(I.)  The  relay,  or  the  relay  magnet  (Fig.  430),  consists  of  an 
electro-magnet,  an  armature,  a  lever,  and  a  spring,  the  same  as  in 
Fig.  427,  except  that  the  electro-magnet  is  horizontal,  and  the 
other  parts  correspond  in  position.  The  tension  of  the  spring  s 


is  regulated  by  the  screw  and  milled  head  h,  and  M  is  adjusted 
by  a  similar  screw  (between  W'  and  W"  in  the  figure),  which 
slides  it  along  the  grooved  way  X.  One  end  of  the  coil  wire 
passes  out  through  W '  to  W  of  Fig.  429.  The  other  end  con- 
nects at  W"  with  one  pole  of  the  battery  if  it  is  at  S  (Fig.  428), 
with  the  earth  if  it  is  at  S1,  or  with  the  line  wire  to  the  next 
station  if  it  is  at  s. 

The  reason  for  introducing  the  relay  is  this:  The  current 
from  the  preceding  station  has  become  too  feeble  to  cause  inden- 


474  DYNAMICAL    ELECTRICITY. 

tation  of  the  paper  by  the  style,  and  thus  make  a  visible  record,  or 
even  to  produce  a  distinct  sound  of  the  armature  upon  the  magnet 
for  reading  messages  by  the  ear.  The  relay  is  therefore  contrived 
for  employing  this  feeble  current  to  close  and  open  the  circuit  of 
a  local  battery,  whose  current  is  powerful  enough  to  deliver  mes- 
sages in  either  form,  or  even  in  both  forms  at  once.  All  which 
the  weak  current  of  the  distant  battery  has  to  do  is  to  cause  the 
armature  A  to  move  toward  the  magnet  M  till  the  top  of  L 
touches  the  screw  N,  and  thus  closes  the  circuit  of  the  local  bat- 
tery. When  the  current  ceases,  a  delicate  spring,  s,  draws  L 
back  from  contact  with  N,  and  breaks  the  circuit  of  the  local 
battery.  By  the  adjustments  above  described,  the  distance 
through  which  L  moves,  and  the  force  of  the  spring  s,  may  be 
made  as  small  as  the  operator  pleases. 

(2.)  The  second  part  of  the  indicator  is  either  a  register,  or  a 
sounder,  according  as  messages  are  to  be  addressed  to  the  eye  or 
to  the  ear.  The  register  (Fig.  427)  has  been  already  described  ; 
the  current  of  the  local  battery  close  at  hand  has  force  enough  to 
cause  visible  indentations  in  the  paper  whenever  the  lever  is  drawn 
to  the  magnet ;  and  this  record  can  be  read  at  any  time  subse- 
quently. Within  a  few  years  a  modified  form  of  the  register  has 
come  into  use,  and  is  called  the  sounder.  In  this  the  end  of  the 
lever  L'  (Fig.  431),  instead  of  being  furnished  with  a  style,  is 
made  to  strike  against  the  two  screws,  Nr,  0'.  The  downward 


click  is  a  little  louder  than  the  upward  one,  and  so  the  beginning 
and  end  of  each  dot  or  dash  are  distinguished  from  each  other. 
Many  operators  learn  from  the  first  to  read  by  the  ear,  and  have 
never  used  a  register. 

Whether  a  register  or  a  sounder  is  employed,  its  coil  wire  is 
entirely  distinct  from  the  line  wire,  except  on  very  short  circuits, 
and  belongs  only  to  the  local  battery.  The  circuit  of  this  battery 
may  be  traced  (Figs.  430,  431)  from  the  positive  pole  through 


RESISTANCE    COILS.  475 

I,  L,  N,  n,  o o,  ri,  the  coil  of  the  sounder,  and  n",  to  the 

negative  pole.  The  binding  screws,  I,  n,  ri,  n",  are  connected 
with  their  respective  levers,  or  contact  screws,  by  insulated  wires 
concealed  in  the  bases.  When  A  is  attracted  by  M9  L  touches  N9 
and  the  circuit  is  closed  ;  when  it  is  withdrawn  by  s,  the  circuit 
is  opened,  because  0  is  insulated.  Hence  the  motions  of  L  and 
L'  are  simultaneous. 

Since  the  relay  is  always  in  the  main  circuit,  it  communicates, 
by  means  of  the  local  current,  to  the  operator  at  whose  station  it 
is,  all  the  messages  sent  between  any  two  stations  on  the  line, 
including  those  which  he  himself  sends.  Hence,  if  his  own 
indicator  does  not  operate  while  he  is  at  work,  he  knows  that  his 
message  is  not  passing  over  the  line,  owing  to  some  break  in  the 
circuit. 

748.  Repeaters. — On  a  well  insulated  wire  the  weakness  of 
the  current  at  the  distance  of  a  few  miles  from  the  battery  is 
mainly  due  to  the  resistance  of  the  wire.     The  nature  of  this 
resistance  is  unknown,  but  it  is  subject  to  the  same  law  as  the 
friction  of  a  fluid  along  the  interior  of  a  tube  ;  it  varies  directly 
as  the  length,  and  inversely  as  the  diameter.     Hence  telegraph 
wires  of  considerable  thickness  are  employed,  and  even  then,  after 
a  certain  number  of  miles,  varying  according  to  strength  of  bat- 
tery, insulation,  etc.,  the  current  will  not  work  even  a  relay. 

A  battery  equivalent  to  25  Daniel  1  cells  will  work  a  relay 
through  not  more  than  about  100  miles  of  ordinary  telegraph 
wire ;  before  reaching  that  point,  therefore,  the  wire  is  allowed 
to  pass  into  the  ground,  and  so  complete  the  circuit. 

To  pass  a  message  beyond  the  place  at  which  the  current  will 
only  work  a  relay,  N'  and  L'  are  made  parts  of  a  new  circuit 
called  a  repeater,  which  is  closed  and  opened  simultaneously  with 
the  preceding  one  by  the  motions  of  L',  just  as  a  local  circuit  is 
worked  by  L  (Art.  747). 

On  the  28th  of  February,  1868,  signals  were  sent  through 
from  Cambridge,  Mass.,  to  San  Francisco,  by  the  employment  of 
thirteen  repeaters.  The  time  occupied  by  the  signals  in  going 
and  returning  (making  about  7,000  miles)  was  three-tenths  of  a 
second— allowance  being  made  for  the  coil  wires  of  the  electro- 
magnets through  which  the  current  passed. 

749.  Resistance  Coils. — In  order  to  explain,   in  a  very 
general  way,  the  principle  of  Duplex  Telegraphy  it  is  necessary 
first  to  consider  the  practical  measurement  and  distribution  of 
resistances.     The  practical  unit  of  resistance  has  been  defined  to 
be  the  ohm  (Art.  690),  and  wire  coils  of  resistance  of  a  definite 


476 


DYNAMICAL    ELECTRICITY. 


FIG.  432. 


number  of  ohms  are  furnished  by  makers  of  electrical  apparatus. 
Fine  German-silver  wire  is  generally  used  because  of  its  constancy 
under  changes  of  temperature  (Art.  690)  ;  and  to  prevent  induc- 
tion in  the  coil  itself  the  insulated  wire  is  doubled  before  being 
wound  so  that  the  two  parts  of  the  wire  carry  opposite  currents, 
thus  neutralizing  each  other's  effect. 

In  Fig.  432  are  represented,  first  in  elevation  and  second  in 
plan,  two  such  coils  of  a  set.  E,  H,  and  M  are  brass  plates,  each 

insulated  from  all  others  by  being  bed- 
ded in  a  plate  of  vulcanite,  A,  B.  One 
end  of  the  first  coil  is  attached  to  E  at 
a,  the  other  end  of  the  same  coil  being 
attached  to  H  at  b.  The  attachments 
of  the  second  coil  are  evident  from  the 
figure.  Two  brass  plugs,  1  and  2,  may 

m be  inserted  in  the  holes  at  the  ends  of 

^4  the  brass  plates,  thus  making  electrical 

connection  between  the  plates  E,  H,  and 

M.  Let  one  wire  of  a  battery  be  attached  to  M  and  the  other, 
after  passing  through  a  galvanometer,  be  attached  to  E,  then  if 
the  plugs  be  removed  the  current  will  flow  from  M  through  the 
coil  to  ff,  and  thence  through  the  connecting  coil  to  E,  and  so 
through  the  galvanometer  to 
the  battery  again,  producing  a 
certain  deflection  of  the  needle ; 
if  now  the  plugs  be  inserted 
the  current  will  flow  from  M 
through  the  plug  2  to  H  and 
thence  through  plug  1  to  E, 
and  the  lessened  resistance, 
due  to  the  removal  of  the  two 
coils  from  the  circuit,  will  be 
manifested  by  the  greater  de- 
flection of  the  needle. 

A  set  of  Eesi stance  Coils  is 
shown  in  Fig.  433.  A  box  may 
contain  coils  of  the  following 

convenient  resistances  in  ohms  :  1,  2,  2,  5,  10,  20,  20,  50,  100,  200, 
200,  500,  and  so  on  to  any  desired  total. 

750.  Wheatstone's  Bridge.— If  a  current  flows  through  a 
wire  of  uniform  resistance,  the  potential  at  the  end  connected  with 
the  positive  pole  of  the  battery  will  be  highest,  and  the  fall  in 
potential  will  be  uniform  to  the  negative  pole  where  it  will  be 
least,  so  that  when  the  current  has  traversed  one-third  of  the 


FIG.  433. 


MEASUREMENTS    OF    RESISTANCE. 


477 


wire,  that  is  to  say,  one-third  of  the  resistance,  the  fall  of  potential 
will  have  been  one-third  of  the  total  fall ;  in  other  words,  the  fall  of 
potential  is  proportional  to  the  resistance  overcome.  A  constant  cur- 
rent arriving  at  A  (Fig.  434),  will  divide,  one  portion  going  through 
the  wires  m  and  n  to  (7,  and  the  other  through  o  and  p  to  C,  at 
which  point  the  currents 

again  unite  and  form  a  FlG-  434- 

single  current  as  before.  B 

At  A  the  potential  of  the 
current  is  less  than  when 
it  left  the  battery  pole, 
and  the  fall  continues 
through  both  m  and  n  to 
(7,  and  also  through  o  and 
p  to  (7,  at  which  point 
the  two  potentials  are 

equal,  the  fall  in  these  arms,  as  they  are  called,  being  propor- 
tional to  the  resistances  which  they  offer.  If  the  resistance  of 
the  arm  m  be  80  ohms,  of  the  arm  n  20  ohms,  of  the  arm  o  4 
ohms,  and  of  p  1  ohm,  then,  since  the  divided  currents  have  the 
same  potential  on  leaving  A,  and  the  same  lower  potential  on 
arriving  at  (7,  it  follows  that  £  of  the  total  fall  will  have  taken 
place  at  B  on  the  upper  circuit,  and  also  -J  of  the  same  total  fall 
at  B'  on  the  lower  circuit ;  hence  the  potential  at  B  is  the  same 
as  that  at  B'9  and  a  galvanometer  on  a  branch  connecting  these 
two  points  would  show  that  no  current  flows  between  B  and  B'. 
Kesistances  and  a  galvanometer  thus  arranged  were  used  by 
Wheatstone,  and  the  device  is  commonly  known  as  Wheatstone's 
Bridge  or  Balance. 

751.  Measurements  of  Resistance. — Suppose  we  wish  to 
determine  the  resistance  of  one  mile  of  a  given  wire.  Let  us 
arrange  our  balance  so  that  the  wire  to  be  measured  shall  be  the 
arm  p,  and  suppose  we  have  made  m  =  100  ohms  and  n  =  10 
ohms,  and  that  we  find  it  necessary  to  make  o  =  110  ohms  to 
bring  the  needle  of  G  to  zero ;  then, 

n  x  o 

m  :  n  : :  o  :  p,  or  p  = -, 

which  gives  p  =  *f££-  =  11  ohms. 

To"  measure  the  internal  resistance  of  a  cell,  a  number  of  resist- 
ance coils  and  a  galvanometer  are  included  in  the  circuit.  If  now 
a  second  cell,  in  all  respects  like  the  first,  be  joined  with  the  first 
in  multiple  arc,  the  internal  resistance  will  be  only  one-half  as 
great  as  before,  and  added  resistance  must  be  introduced  into  the 
circuit  to  bring  the  needle  back  to  its  former  reading.  Call  the 


478 


DYNAMICAL    ELECTRICITY. 


electromotive  force  of  the  cell  E,  its  resistance  Jit  the  resistance 
of  the  external  circuit  in  the  first  case  r  and  in  second  r',  and 
then,  since  the  galvanometer  showed  the  strength  of  the  current 
to  be  the  same  in  both  cases  we  have,  according  to  Ohm's  Law, 

E  _E_ 

R  +  r-i  JT+r" 
from  which  we  find 

R  =  2  (r'  —  r). 

To  compare  the  electromotive  force  of  a  given  cell  with  that 
of  a  standard  cell,  introduce  into  a  circuit  resistance  coils  and  a 
galvanometer  ;  first  pass  a  current  through  from  the  standard  cell, 
and  so  arrange  the  resistance  that  the  galvanometer  shall  read  any 
convenient  number  of  degrees,  which  call  d  ;  then  add  resistance, 
say  100  ohms,  so  that  the  reading  shall  fall  to  d'  ;  now  substitute 
for  the  standard  cell  that  whose  relative  electromotive  force  we 
wish  to  determine,  and  arrange  the  resistances  so  that  the  reading 
may  be  d,  as  before,  and  then  add  enough  resistance,  say  190 
ohms,  to  bring  the  reading  down  to  d'  again.  Now  since  the 
resistances  which  will  reduce  the  galvanometer  readings  by  the 
same  amount  are  proportional  to  the  electromotive  forces  we  have, 
calling  E  the  electromotive  force  of  the  standard  cell  and  E'  that 
of  the  other, 

E  :  E'  :  :  100  :  190,  or  E'  = 


752.  Duplex   Telegraphy.—  Let  Fig.   435   represent  two 
stations  connected  by  the  line  wire  L  L'.     C  L  R  is  a  Wheatstone 


FIG.  435. 


Bridge,  modified  to  suit  the  conditions  of  the  case,  /  the  register 
or  sounder,  R  resistance  coils,  Tc  a  key  working  upon  the  center 
and  having  forward  and  back  contacts  at  a  and  c,  b  the  battery, 
and  E  the  earth  connections.  The  same  letters,  accented,  repre- 
sent like  parts  at  the  second  station.  When  not  in  use  the  keys 
make  back  contact  by  the  action  of  a  spring.  The  ratio  of  the 
resistance  0  R  and  R  E$  is  made  equal  to  that  of  C  L  and  the 
line  wire  L  L',  including  the  back  contact  earth  connection  at  the 


ATLANTIC    TELEGRAPH    CABLE.  479 

second  station.  If  now,  a!  being  closed,  a  be  closed,  a  current 
will  flow  through  a  and  k  to  C,  where  it  will  divide,  one  part 
going  to  earth  through  R  and  E3,  and  the  other  through  L  L'. 
As  the  potentials  were  made  equal  at  L  and  R,  no  current  will 
pass  through  the  indicator  /;  that  part  of  the  current  which  flows 
through  L  L'  divides  at  L',  part  going  through  G'  k'  a'  to  E", 
and  part  through  /'  (giving  signal)  and  R'  to  E.  Thus  the 
closing  of  a  gives  a  signal  at  /'  but  none  at  /. 

If  now  the  second  operator  should  close  his  key  while  a  was 
closed,  a  current  from  V  would  flow  through  c'  and  k'  to  C',  where 
it  would  divide,  part  going  to  earth  through  R'  and  E'  (joining 
the  current  already  flowing  through  from  L  L'),  and  part  would 
flow  to  L'  and  oppose  the  current  from  the  other  station ;  this 
opposing  current  will  have  the  same  effect  as  increased  resistance 
in  the  line  wire  L  L',  and  hence  the  balance  C  L  R  will  be  dis- 
turbed, the  potential  of  L  rising  above  that  of  R,  and  resulting  in 
a  current  from  L  through  /to  R,  giving  a  signal  at  /.  Thus  the 
register  at  each  station  will  respond  to  the  key  of  the  other,  and 
only  to  that,  whether  one  or  both  operators  be  signalling. 

The  above  explanation  of  the  principle  of  this  particular  mode 
of  sending  simultaneous  messages  in  opposite  directions  on  a 
single  wire,  does  not  pretend  to  describe  the  actual  arrangement 
of  wires  or  earths  in  use.  For  a  full  description  of  the  various 
modes  of  duplex  and  quadruplex  telegraphy  the  student  is  referred 
to  works  on  practical  telegraphy. 

753.  Atlantic  Telegraph  Cable.— This  cable  stretches  a 
distance  of  3,500  miles,  and  from  the  nature  of  the  case  is  a  con- 
tinuous wire,  so  that  it  cannot  be  advantageously  worked  by  the 
Morse  apparatus.  The  indicator  employed  is  a  sensitive  galva- 
nometer needle  which  is  made  to  oscillate  on  opposite  sides  of  the 
zoro  point  by  the  passage  through  it  of  currents  in  opposite  direc- 
tions. But  to  reverse  the  direction  of  the  current  throughout  the 
whole  length  of  the  cable  is  a  slow  process.  For  tlie  cable  is  an 
immeme  Ley  den  jar,  the  surface  of  the  copper  wire  (amounting  to 
425,000  sq.  feet)  answering  to  the  inner  coating,  the  water  of  the 
ocean  to  the  outer,  and  the  gutta-percha  between  the  two  to  the 
glass  of  an  ordinary  jar.  A  current  passing  into  it  is  therefore  de- 
tained by  electricity  of  the  contrary  kind  induced  in  the  water,  and 
no  effect  will  be  produced  at  the  further  end  until  it  is  charged. 

This  very  circumstance,  at  first  considered  a  misfortune,  is 
now  taken  advantage  of  in  a  very  simple  and  ingenious  manner  to 
facilitate  the  transmission  of  signals.  The  current  is  allowed  to 
pass  into  the  cable  till  it  is  charged— then,  without  breaking  the 
circuit,  by  depressing  a  key  for  an  instant,  a  connection  is  made 


480  DYNAMICAL    ELECTRICITY. 

between  it  and  a  wire  running  out  into  the  sea ;  that  is,  between 
the  inner  and  outer  coatings.  Hiis  partially  discharges  it,  and 
the  needle  at  the  other  end  is  deflected.  When  the  key  is  raised 
the  discharge  ceases,  the  current  flows  on  as  before,  and  the  needle 
is  deflected  in  the  opposite  direction. 

It  is  said  that  after  this  plan  was  adopted,  twenty  words  could 
be  sent  through  the  cable  per  minute,  whereas  only  four  per 
minute  could  be  sent  before.  The  greatest  speed  thus  far  attained 
on  land  wires  is  believed  to  have  been  the  transmission  in  one 
instance  of  1,352  words  in  thirty  minutes  between  New  York  and 
Philadelphia  in  1868. 

754.  Fire- Alarm  Telegraph.  —  Kecurring  again  to  the 
standard  (Fig.  425),  the  end  E  may  be  so  connected  with  ma- 
chinery as  to  cause  the  striking  of  a  bell  in  a  distant  tower  when- 
ever the  circuit  is  closed  at  n  p.    In  our  large  cities  boxes  are 
placed  at  convenient  points,  each  containing  a  crank,  or  lever,  by 
which  the  circuit  may  be  closed  and  the  fire-bell  rung. 

Fig.  436  represents  a  device  by  which  the  number  32  may  be 
signalled  to  a  central  station  by  setting 
FIG.  436.  free  the  wheel  so  that  it  may  turn  in  the 

direction  of  the  arrow.  Strips,  a  a  a  and 
a'  a',  of  insulating  material  are  fastened 
upon  the  rim  of  a  metallic  wheel ;  the 
two  wires,  b  and  c,  of  the  general  circuit, 
press  upon  the  metallic  rim  and  the  cir- 
cuit is  closed.  If  the  wheel  "be  turned 
the  three  strips  a  a  a,  passing  under  b 
and  c,  cause  three  successive  interruptions 
of  the  circuit,  followed  after  a  longer 
interval  by  the  two  breaks  due  to  a'  a'. 
Thus,  by  previously  arranged  signals, 

the  locality  of  the  fire  is  immediately  made  known  at  the  various 

engine-houses. 

755.  Chronograph. — This  is  used  in  observatories  for  record- 
ing the  passage  of  stars  across  the  meridian.     Imagine  the  circuit 
of  Fig.  427  to  be  closed  and  instantly  broken  again,  by  a  clock 
pendulum  at  the  end  of  every  second.     As  the  paper,  R  R,  moves 
uniformly,  dots  are  made  on  it  at  equal  distances  from  each  other, 
each  of  which  distances,  therefore,  represents  one  second.     The 
observer  has  a  key,  by  which  also  he  closes  the  circuit  for  an 
instant  when  a  certain  star  passes  the  meridian.     The  dot  thus 
made  shows,  by  its  situation  between  the  two  nearest  second  dots, 
at  what  fraction  of  the  second  the  transit  occurred. 

In  practice,  however,  the  record  is  more  conveniently  made  on 


CHRONOGRAPH.  481 

a  large  sheet  of  paper,  which  is  wrapped  tightly  around  a  cylinder. 
The  clock-work,  which  revolves  the  cylinder,  also  moves  the  re- 
cording pen  in  a  line  parallel  to  its  axis.  By  these  two  motions, 
a  spiral  ink-line  is  traced  on  the  paper.  At  the  end  of  every  beat 
of  the  observatory  clock,  the  closing  of  the  circuit  gives  the  pen  a 
momentary  lateral  movement,  by  which  a  slight  notch  is  made  in 
the  line.  A  similar  notch  is  made  by  the  touch  of  the  key,  when 
the  observer  perceives  the  star  on  the  meridian  wire  of  the  tele- 
scope. Fig.  437  represents  a  portion  of  the  sheet  after  its  removal 

FIG.  437. 


from  the  cylinder ;  a,  b,  c,  d,  &c.,  are  the  second  marks  ;  a?,  y,  z, 
&c.,  are  transit  records.  The  ratio  my  :  m  n  shows  what  fraction 
of  the  second  m  n  has  elapsed  when  the  transit  y  occurs. 


APPENDIX, 


APPLICATIONS  OF  THE  CALCULUS. 

I.  FALL  OP  BODIES. 

1.  Differential  Equations  for  Force  and  Motion.— These 

are  three  in  number,  as  follows : 

ds 

I.       v  —  TV 
d  t 

9  f-^-CLS 

J  '    dt      df 
3.  fds  —  v  dv. 

These  equations  are  readily  derived  from  the  elementary  prin- 
ciples of  mechanics.  In  Art.  G  we  have  v  —  .  Reducing  the 

t 

numerator  and  denominator  to  infinitesimals,  v  remains  finite,  and 

d s 
the  equation  becomes  v  =  -7-  ;  which  is  Equation  1st.    Therefore, 

Ci   t 

if  the  space  described  by  a  body  is  regarded  as  a  function  of  the 
time,  the  first  differential  coefficient  expresses  the  velocity. 

Again  (Art.  12),  /=  -,  where  /represents  a  constant  force. 
Making  velocity  and  time  infinitely  small,  we  get  the  intensity  of 
the  momentary  force,  /—  -,->  But,  by  Equation  1st,  v  =-?-', 

d*  s 
/./—  -y^;  which  is  Equation  2d.    Hence  we  learn  that  the/rrf 

Ct  v 

differential  coefficient  of  the  velocity  as  a  function  of  the  time,  or 
the  second  differential  coefficient  of  the  space  as  a  function  of  the 
time,  expresses  the  force. 

Equation  3d  is  obtained  by  multiplying  the  1st  and  2d  cross- 
wise, and  removing  the  common  denominator. 

We  proceed  to  apply  these  equations  to  the  preparation  of  for- 
mulae for  falling  bodies. 

2.  Bodies  falling  through  Small  Distances  near  the 
Earth's  Surface. — In  this  case,  let  the  accelerating  force,  which 


484  APPENDIX. 

is  considered  constant,  be  called  g.    Then,  by  Eq.  2,  g  —  -T-  , .-.  d  r 

u  t 

—  gdt.     Integrating,  we  have  v  —  g  t  +  C.     But,  since   v  —  0 
when  t  =  0,  .*.  v  =  g  t,  and  t  =  -,  as  in  Art.  27. 

Again,  substituting  g  t  for  v  in  Eq.  1,  ds=gtdt;  and  by 
integration,  s  =  £gf  +C;  but  C  =  0,  for  the  same  reason  as  be- 

/2s 
fore ;  .-.  s  =  ^  g  f ,  and  t  =  \  — . 

Once  more,  equating  the  two  foregoing  values  of  t,  we  have 

va 

v  =  V2  g  5,  and  s  —  ~-. 
*9 

If,  in  the  equation,  s  —  %g  t9,  v  be  substituted  for  #  t,  we  have 
s  =  £  v  tf,  or  v  tf  =  2  s ;  that  is,  the  acquired  velocity  multiplied  by 
the  time  of  fall  gives  a  space  twice  as  great  as  that  fallen  through 
(Art  21). 

3.    Bodies  falling  through  Great  Distances,        FIG.  1. 
so  that  Gravity  is  Variable,  according  to  the 
Law  in  Art.  16. 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  1),  to- 
ward the  centre  (7.  Let  A  C '=  a;  B  C—x\  D  C=r, 
the  radius  of  the  earth. 

The  force /at  B,  is  found  by  the  principle,  Art.  16, 

x*:r9::g:f=gr*~,  =  ffr'x-\ 


4.  To  find  the  Acquired  Velocity.  —  Substitute  g  r*  x~*  for 
f,  and  a  —  x  for  s,  in  Equation  3d,  and  we  have  g  r*  x~*  .d(a—x) 
—  vdv\  /.by  integration  %  v*  =  f  —  g  r*  x~*  d  x  =  g  ra  x~l  +  (7. 
But  t;  =  0,  when  x  —  a  ;  /.  C  =  —  g  r*a~*  ;  and 


ax 
.     S 


(          ax 

This  is  the  general  formula  for  the  acquired  velocity.     If  the 
body  falls  to  the  earth,  x  =  r,  and  the  formula  becomes 


FALL    OF    BODIES.  485 

Again,  if  the  body  falls  to  the  earth  through  so  small  a  space 

M 

that  -  may  be  regarded  as  a  unit,  the  formula  reduces  to 


the  same  as  obtained  by  other  methods. 

If  a  body  falls  to  the  earth  from  an  infinite  distance,  it  does 
not  acquire  an  infinite  velocity.  For  then,  as  we  may  put  a  for 
a  —  r, 


V  = 


(2 .  32 1 .  3956  .  5280)7'  feet  =  6.95  miles. 

Therefore,  the  greatest  possible  velocity  acquired  in  falling  to 
the  earth  is  less  than  seven  miles  ;  and  a  body  projected  upward 
with  that  velocity  would  never  return. 

5.  To  find  the  Time  of  Falling. — From  equation  first  we 

^  a  \%  o  r*(tt #M 

obtain  d  t — —  ;  in  this,  substitute  d(a — x)  for  d  s,  and  — —     — — - 

for  v,  as  found  in  the  preceding  article ;  then 

—  x)  _  /    a    y    —  x2  d  x  a 
~^  ~  V27?J 


(a  -  x) 

/.  by  integration  t  —  ( ~ j   .  J  —  #2  d  x  (a  —  *)""*. 

By  the  formula  in  the  calculus  for  reducing  the  index  of  x  we 
obtain 

-  x^d  x(a-  a?)-*  =(ax-  x^  -  |  vers'1  (^)  +  C. 
Now,  when  t  =  0,  x  —  a ;  .*.  C  =  -^  ; 


6.  Bodies  falling  within  the  Earth  (sup-  FIG.  2. 

posed  to  be  of  uniform  density),  where 
Gravity  Varies  as  the  Distance  from  the 
Centre. — 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  2) ; 
and  let  D  C  =  r,  A  C  =  a,  and  B  C  =  x.    Then 


r  :  x  ::  g  if  —  -x  —  force  at  B. 


486  APPENDIX. 


To  find  the  velocity  acquired.  —  By  Eq.  3d, 

vdv  =fds;  .'.  vd  v  —  ^x.d  (a  —  x)  =—  ^—   -; 

ax* 
.%  ^  v*=  —  ~~-  +  C\  but  v  —  0  when  x  =  a; 


If  the  body  falls  from  the  surface  to  the  centre,  x  —  0,  and 

this  formula  becomes  r  =  (gr)%  =  (32£  x  3956  x  5280)^  =  25,904 
feet  per  second. 

To  find  the  time  of  falling.  —  By  Equation  1st,  and  substitu- 

...  ds      d(a—x]  dx  —dx 

tions,  we  obtain  d  t  —  —  =  —  -  ---  =  --  = 

" 


When  <  =  0,  »  =  <*,-  =  1,  and  the  arc,  whose  cosine  is  1  =  0: 
a 


=  1-1    x  cos  *  -. 
a 


r\$ 


If  the  body  falls  to  the  centre,  x  =  0,  and  t  =  (-)     x  ~  ;  in 

\ff'        & 

which  a  does  not  appear  at  all  ;  so  that  the  time  of  falling  to  the 
centre  from  any  point  within  the  surface  is  the  same  ;  and  equals 


x  1.570796  in  seconds,  or  21m.  5.8s. 


TL  CENTRE  OF  GRAVITY.  , 

7.  Principle  of  Moments. — In  order  to  apply  the  processes 
of  the  calculus  to  the  determination  of  the  centre  of  gravity,  the 
principle  is  used,  which  was  proved  (Art.  78),  that  if  every  par- 
ticle of  a  body  be  multiplied  by  its  distance  from  a  plane,  and 
the  sum  of  the  products  be  divided  by  the  sum  of  the  particles, 
the  quotient  is  the  distance  of  the  common  centre  from  the  same 
plane.     The  product  of  any  particle  or  body  by  its  distance  from 
the  plane,  is  called  its  moment  with  respect  to  that  plane. 

8.  General  Formulae. — Let  B  AC  (Fig.  3)  be  any  symmetri- 
cal curve,  having  A  X  for  its  axis  of  abscissas,  and  A  Y,  at  right 


APPLICATION    OF    FORMULAS. 


487 


FIG.  3. 


angles  to  it,  for  its  axis  of  ordinates.  It  is  obvious  that  the 
centre  of  gravity  of  the  line  B  A  C,  of  the  area  B  A  C,  of 
the  solid  of  revolution  around  the  axis 
A  X,  and  of  the  surface  of  the  same 
solid,  are  all  situated  on  A  X,  on  ac- 
count of  the  symmetry  of  the  figure. 
It  is  proposed  to  find  the  formula  for 
the  distance  of  the  centre  from  A  Y, 
in  each  of  these  cases.  Let  G  in  every 
instance  represent  the  distance  of  the 
genera]  centre  of  gravity  from  the  axis 

A  Y,  or  the  plane  A  Y,  at  right  angles  to  A  X.  The  distance  G 
would  plainly  be  the  same  for  the  half  figure  B  A  D,  as  for  the 
whole  B  A  (7;  expressions  may  therefore  be  obtained  for  either, 
according  to  convenience. 

1.  The  line  A  B. — Let  x  be  the  abscissa,  and  y  the  ordinate ; 

then  (dx*  +  dy*yi  is  the  differential  of  the  line  A  B.  For  brevity, 
let  s  =  the  line,  and  d  s  its  differential.  If  we  now  multiply  this 
differential  by  its  distance  from  A  Y,  x  d  s  is  the  moment  of  a 
minute  portion  of  the  line ;  and  the  integral  of  it,  /  x  ds,  is  the 
moment  of  the  whole.  Dividing  this  by  the  line  itself,  i.  e.  by  s, 

we  have  ' for  the  distance  G. 

s 

2.  TJie  area  B  A  D.— The  differential  of  the  area  is  y  dx ;  the 
differential  of  its  moment  is  x  y  d  x ;   hence  the  moment  itself  is 

C  T  Qi  /I  y 

f  xy  dx:  and  the  distance  G  = * . 

area 

3.  TJie  solid  of  revolution. — The  differential  of  the  solid,  gen- 
erated by  the  revolution  of  A  B  on  A  X,  is  TT  y*d  x ;  the  differen- 
tial of  its  moment  is  n  x  y'*d  x ;  and  the  moment  is  /  Ttxy*dx\ 

„      firx  ifd  x 

hence  the  distance  G  = ,-^ • 

solid 

4.  The  surface  of  revolution. — The  differential  of  the  surface  is 
2  TT  y  d  s ;  the  differential  of  its  moment  is  2  n  x  y  d  s  ;  and  there- 
fore the  moment  isfZ-rrxyds',  and  the  distance  G  —     surfa^e    " 

9.  Application  of  Formulae.— We  proceed  to  determine 
the  centre  of  gravity  in  a  few  cases  by  the  aid  of  these  formulae : 

1.  A  straight  line. — Imagine  the  line  placed  on  A  X,  with  one 
extremity  at  the  origin  A.  The  moment  of  a  minute  part  of  it  is 
xdx,  and  that  of  the  whole  is  /  x  d  x,  while  the  length  of  the 

whole  is  x ;  /.  G  =  ^^-  =  i£±J2  =  £  z,  as  it  evidently  should 


vers 


488  APPENDIX. 

be.  In  all  the  cases  considered  here,  C=  0,  because  the  function 
vanishes  when  x  does. 

2.  The  arc  of  a  circle. — By  formula  1st  we  have6r  = •  but 

s 

d's  =  (d  x*+  dy*y  ;  by  the  equation  of  the  circle,  y1  —  2  a  x  —  x* ; 

,                    .  j          -.   „     (a  —  xYdx*      (a  —  xYdx* 
.'.  ydy  =  (a-x)  dx-} .-.  dy*=  v -£—   =  V^  -^-; 

w 

m 
t  J3      3          J7       1\  "A 

fxds_    f*x            adx         _  a  r       xdx        __a  ( 
~^     ~       s  *  (Saz-ar^    '  s      (2az-o;a)i  ~  *  * 
—  (2  a  a;  —  x*)1*  (.  =  -  (s  —y)  —  a =  a  — — ,  if  the  arc  is  dou- 

I  S  S  t 

bled  and  called  tf ,  and  c  (chord)  put  for  2  y.  As  a y-  is  the  dis- 
tance from  the  origin  A,  and  a  =  radius  of  the  arc ;  /.  the  distance 
from  the  centre  of  the  circle  to  the  centre  of  gravity  of  the  arc, 

is  — ,  which  is  a  fourth  proportional  to  the  arc,  the  chord,  and  the 
t 

radius. 

When  the  arc  is  a  semi-circumference,,  c  =  2  a,  and  t  =  n  a ; 
/.  the  distance  of  the  centre  of  gravity  of  a  semi-circumference 

from  the  centre  of  the  circle  is  — . 

7T 

3.  TJie  area  of  a  circular  sector. — Suppose  the  given  sector  to 
be  divided  into  an  infinite  number  of  sectors ;  then  each  may  be 
considered  a  triangle,  and  its  centre  of  gravity  therefore  distant 

from  the  centre  of  the  circle  by  the  line  -^-.    Hence  the  centres  of 

o 

gravity  of  all  the  sectors  lie  in  a  circular  arc,  whose  radius  is  —  ; 

o 

so  that  the  centre  of  gravity  of  the  whole  sector  coincides  with 
the  centre  of  gravity  of  that  arc.  The  distance  of  the  centre  of 
gravity  of  the  arc  from  the  centre  of  the  circle,  by  the  preceding 

O  O  O  O  fj  f, 

case, is^«  x  -c-+-~t  =  -=-;->  which  is  therefore  the  distance  of 
o          o          o  o  t 

the  centre  of  gravity  of  the  sector  from  the  centre  of  the  circle. 

When  the  sector  is  a  semicircle  the  distance  becomes  — 5 

o  na 


CENTRE    OF    GRAVITY.  489 

4.  The  area  of  a  parabola.  —  The  equation  of  the  curve  is 

/  =  px,  my=p^x$; 
therefore  the  formula  2  for  moment, 

fxydx  =  fp*x%dx  =  fj^  &  (  +  <7=  Q); 
but  the  area  of  the  half  parabola  =  f  p*  x~*  ; 

.-.  G  =  |  pi  x%  +  lp^  x%  =  |  x. 

To  find  the  distance  of  the  centre  of  gravity  of  the  semi-parab- 
ola from  the  axis  A  X,  proceed  as  follows  :  The  differential  of  the 
area,  as  before,  equals  y  d  x  ;  and  the  distance  of  its  centre  from 
A  X  is  ^  y.  Therefore  its  moment  with  respect  to  AX  is  ^  ya  d  x 
=  5  p  x  d  x  ;  and  the  moment  of  the  whole  is  /£  p  x  d  x  = 
:.  the  distance  of  the  centre  from 


A  X  =  ±px*  +  Jj>3    =  Ip    x*  =  jy. 
5.  The  area  of  a  circular  segment.  —  The  equation  of  the  circle 
is,  y  =  (2  a  x  —  x*y.    Therefore  (formula  2), 

fxydx  =fx  (2  ax  -x^dx. 

Add  and  subtract  a  (2  ax  —  a;')2  d  x,  and  it  becomes 

fa  (2ax  -  x^dx  -f(a  -  x)  (2ax  —  x*)?dx  = 


_  (tax-*)* 

3  area  A  B  D' 

4  a 
When  x  =  a,  G  —  a  —  s-  ;  and  the  distance  of  the  centre  of 

O  7T 

4a 
gravity  of  a  semicircle  from  the  centre  of  the  circle  =  5-  .    When 

x  =  2  a,  G  —  a,  as  it  plainly  should  be. 

6.  A  spherical  segment.  —  The  equation  of  the  circle  is  y'  = 
2  a  x  —  a?.    Therefore  (formula  3), 

nx*d 

Sax-3x* 


When  x=a,  G  =  f  a;  that  is,  the  centre  of  gravity  of  a  hem- 
isphere is  |  of  radius  from  the  surface,  or  |  of  radius  from  the 
centre  of  the  sphere.  If  x  =  2  a,  G  =  a. 

7.  A  right  cone.—  In  this  case  A  B  (Fig.  3),  is  a  straight  line, 
and  its  equation  is  y  =  a  x,  where  a  is  any  constant 


490 


APPENDIX. 


.-.  y  -  aV;  /.  firxtfdx  =  fira'tfdx  =    flV  ;  .-.  0  =  f-=f 

4:  ~"^a  X 


Hence  the  centre  of  gravity  of  a  cone  is  three-fourths  of  the  axis 
from  the  vertex.     See  Art.  75. 

8.  TJie  convex  surface  of  a  right  cone.  —  The  equation  is 

y  =  ax;  .'.  dy*  =  tfdx*;  and  (dx*  +  rfyf)i  =  (aa  +  I)*dx. 
Therefore  (formula  4), 

a1  +  1)* 


=  the  moment  of  the  surface.     The  surface  itself, 


a3  +1)T 

The  centre  of  gravity  of  the  convex  surface  of  a  right  cone  is  on 
the  axis,  at  a  distance  equal  to  two-thirds  of  its  length  from  the 
vertex. 

III.   CENTRE  OF  OSCILLATION. 

10.  To  find  the  Moment  of  Inertia  of  a  Body  for  any 

S  (m  ra) 
given  Axis. — To  render  the  formula  I  =     .:,,      suitable  to  the 

application  of  the  calculus,  we  have  simply  to  substitute  the  sign 
of  integration  for  S,  and  d  M  for  m,  and  we  have 


Mk 


(1) 


It  is  useful  to  know  how  to  find  the  moment  of  inertia  with  respect 

to  any  axis  by  means  of  the  FIG.  4 

known  moment  with  respect  to 

another  axis  parallel  to  it  and 

passing  through  the  centre  of 

gravity  of  the  body. 

Let  A  Z  (Fig.  4)  be  the  axis 
passing  through  the  centre  of 
gravity  of  the  body  for  which 
the  moment  of  inertia  is  fr*dM, 
and  let  A'  Z'  be  the  axis  paral- 
lel to  it,  for  which  the  moment 
of  inertia,  /r'a  d  M  of  the  same 
mass  M9  is  to  be  determined. 
For  every  particle  m  of  the  body 
the  corresponding  value  of  A  m' 
is  r9  ^  z*  +  y\  In  like  man- 


^ 


CENTRE    OF    OSCILLATION.  4m 

ner,  if  we  denote  the  co-ordinaces  of  A'  by  a  and  0,  and  the  dis- 
tance between  the  axes  by  «,  we  shall  have  a*  =  aa  +  j33.  Now  the 
distance  of  the  particle  m  from  A'  Z'  is  r"  =  (x  —  a)9+  (y  —  /})* 
=  2?  +  ya  +a2  +  &  -%ax-2(3y  =  r»  +  a2-  2az-20v;.v 
/>"  d  M=  JY  d  M  +  d'fdM-  2afxdM-2ftfdM  =  ai  M 


since  .4  ^  passes  through  the  centre  of  gravity  of  the  body.  Hence. 
the  moment  of  inertia  of  a  body  ivith  respect  to  any  axis  is  equal  to 
the  moment  of  inertia  with  respect  to  a  parallel  axis  through  the 
centre  of  gravity,  plus  the  mass  of  the  body  multiplied  by  the  square 
of  the  distance  between  the  two  axes. 

Put  C  =  the  moment  of  inertia  with  respect  to  an  axis  through 
the  centre  of  gravity  ;  then  the  distance  from  the  axis  of  suspen- 
sion to  the  centre  of  oscillation,  the  axes  being  parallel,  will  be 

,_  fl+a'lf 
~~ 


11.  Examples.  — 

1.  Find  the  centre  of  oscillation  of  a  slender  rod  or  straight 
line  suspended  at  any  point. 

Let  a  and  b  be  the  lengths  on  opposite  sides  of  the  axis  of  sus- 
pension, then  by  (1) 

fr*dM_          fr'dr  2  (a3  +  />3)      2  (aa  -  a  b  +  ft1) 

Mk      "  (a  +  b)$(a-b)~3(a*-F)  3  (a  -  b) 

between  the  limits  r  =  ~h  a  and  r  —  —  b. 

If  the  rod  is  suspended  at  its  extremity,  b  =  0,  and  I  =  f  a.    If 
it  is  suspended  at  its  middle  point,  a  —  b  and  I  =  oc  . 

2.  Find  the  centre  of  oscillation  of  an  isosceles  triangle  vibra- 
ting about  an  axis  in  its  own  plane  passing  through  its  vertex. 

Put  b  and  h  for  the  base  and  altitude  of  the  triangle;  t>>en  bjr 


If  the  axis  of  suspension  coincides  with  the  base  of  the  trian- 


gle,  then  I  = p^r — rjr~    ~~  =  2* 

3.  Find  the  centre  of  oscillation  of  a  circle  vibrating  about  an 
axis  in  its  own  plane. 

C=fr'dM=  2fx*ydx  =  2,/V  (R*  -  x*]*dx  = 


492  APPENDIX. 

Taking  this  integral  between  x  —  —  r  and  x  =  -I-  r,  we  have 

n  —        n       — n 
"2*2  4~' 

Substituting  this  value  of  C  in  (3)  we  have 

•f  K  o         -pa 

: \-    (I    TT  11  ™ 


1  = 


aTrfl" 

4.  Find  the  centre  of  oscillation  of  a  cir- 
cle vibrating  about  an  axis  perpendicular 
to  it. 

Let  K  L  (Fig.  5)  be  an  elementary  ring 
whose  radius  is  x  and  whose  breadth  is  d  x ; 
then 

dM=-  rR 


Fro.  5. 


TT  R' 


As  #  +  ~—  is  greater  than  a  +  T-  -,  a  cir- 
cular pendulum  will  vibrate  faster  when  the 
axis  of  suspension  is  in  its  plane,  than  when 
it  is  perpendicular  to  it. 


IV.  CENTRE  OF  HYDROSTATIC  PRESSURE. 

12.  General  Formula. — Let  the  surface  pressed  upon  be 
plane  and  vertical ;  and  let  the  water  level  be  the  plane  of  refer- 
ence. Suppose  the  surface  to  have  a 
symmetrical  form  with  reference  to  a 
vertical  axis,  x,  whose  ordinate  is  y 
(Fig.  6).  A  horizontal  element  of  the 
surface  is  2  y  d  x,  and  (since  the  pres- 
sure varies  as  the  depth)  the  pressure 
on  that  element  2  x  y  d  x.  Hence  the 
whole  pressure  to  the  depth  x  is 
f%xy  dx—  2  /  xy  d  x.  The  mo- 
ment of  the  pressure  on  the  element 

of  surface  is  2  x3  y  d  x ;   and  the  sum  of  all  the  moments  to  the 
same  depth  is  f^x^ydx  —  ^fx^ydx.     Therefore,  putting  p 

for  the  depth  of  the  centre  of  pressure,  p  =  ~ — — =— . 

/  xy  d  x 


CENTRE    OF    HYDROSTATIC    PRESSURE.        493 

13.  Examples. 

I.  A  rectangle.—  Let  its  height  =  h,  and  its  base  =  I  ;  then  2y 
everywhere  equals  b,  and  a  horizontal  element  at  the  depth  x  ia 
b  d  x,  the  pressure  on  it  is  b  x  d  x,  and  the  moment  of  that 
pressure  is  b  x*  d  x  ;  /.  the  depth  of  the  centre  of  pressure  p  = 
/'  bx*dx  ±bx3  +  c  0. 
'~  4  6  a3  +  cr'  e  Pressure  and  area  is  each  zero, 


when  x  is  zero,  c  and  c'  both  disappear,  and  p  =  f  x,  which  for  the 
whole  surface  becomes  p  —  f  h.  That  is,  the  centre  of  pressure  on 
a  vertical  rectangular  surface  reaching  to  the  water  level,  is  two- 
thirds  of  the  distance  from  the  middle  of  the  upper  side  to  the 
middle  of  the  lower. 

2.  A  triangle  whose  vertex  is  at  the  surface  of  the  water,  and  its 
base  horizontal.  —  Let  the  triangle  be  isosceles,  its  height  —  h,  and 

J)  J  hx*^x 

its  base  =  #;  then  h  :  b  :  :  x  :  2  y  =  =-  x.   Therefore^  =  -/rr" 

/!.*«• 

1  x* 

—  -  —  3  =  J  x  ;   and  for  the  whole  height,  |  h. 

3  X 

If  the  triangle  is  not  isosceles,  it  may  be  easily  shown  that  the 
centre  of  pressure  is  on  the  line  joining  the  vertex  and  the  middle 
of  the  base,  at  a  distance  from  the  vertex  equal  to  three-fourths  of 
the  length  of  that  line. 

3.  A   triangle  whose  base  is  at  the  water  level.  —  Then  h  :  b 

:  h  —  x  :2  y  =  b  —  j  x.     Therefore  the  pressure  is  /—  b  x  d  x 

At 

—f  —  =-  x*  d  x,  because  d  x  is  negative.      The  moment  of  the 
fi 

pressure  is  /—  b  x*  d  x  —  /—  -=  x*dx. 

-fbx*dx  +  f\^dx      -  \  x*  4-  -—^ 
Therefore  p  =  •  -- 


_~5_  =  -         .   and,  when  x  =  h,  this  becomes  j  h. 

6  h  x*  —  4  x3         6  h  —  4x    ' 
In  general,  the  centre  of  pressure  is  at  the  middle  of  the  line  join- 

ing the  vertex  and  the  middle  of  the  base. 

i     i 

4.  A  parabola  whose  vertex  is  at  the  surface.—  As  y  =  p2  z2, 


spxdx      fdx_$x?_5  6  ,    f 

therefore  p  =  -  -  -  --  =  -  -  --  =  —  -  —  y  x  >  or  ^  ^  r 

fx^x^dx      fx?dx      \& 
the  whole  area. 


494  APPENDIX. 

5.  A  parabola  whose  base  is  at  the  surface.  —  As  h  —  x  is  the 

depth  of  an  element,  d  x  is  negative,    p  =  ---  -  = 

-  f  (h-x)  x^dx 

f  (h*x?  dx-Zhx*  dx  +  xsdx)  __  j  Aa  x%  -  |  h  x?  +  *  x^ 

f  (h  x?  dx-  z%  dx)  |  h  x*  —  f  z* 

+  i  x9 
--  «  —  j  anc[  when  x  =  n,  the  expression  becomes 


_ 


V.  ANGULAR  RADIUS  OF  THE  PRIMARY  AND  SECONDARY  RAIN- 
BOW AND  THE  HALO. 

14.  The  Primary  Rainbow. — Since  the  primary  bow  is 
formed  by  those  rays  which,  on  emerging  after  one  reflection, 
nuke  the  largest  angle  with  the  incident  rays,  proceed  to  find 
what  angle  of  incidence  will  cause  the  largest  deviation  of  the 
emerging  rays. 

In  Fig.  7,  let  x  —  angle  of  inci-  FIG.  7. 

dence ;  y  —  angle  of  refraction ;  z  — 

angle  of  deviation ;  n  —  index  of  re- 
fraction. Then,  in  the  quadrilateral 
B  D  OK,  DB  K^D  &K'=x-yi 
angle  at  D  =  360  -  2y;  /.  K  =  z  = 


_ 

•  •    ~z —    —        j —  /     —  \* 

dx       dx 
But  sin  x  =  n  sin  y ; 

,  dy        cos  x 

/.  cos  x  d  x  =  n  cos  y  d  i/,  and  -~  =  -      — . 

ax      n  cos  y 

.    ...   , .       4  cos  x 
By  substitution,  -       -  =  2. 
n  cos  y 

.:  2  cos  x  —  «  cos  ^ ;  and  4  cos*  a;  —  n9  cos'  y. 
But  sin2  a;  =  na  sin2  y ; 

>.  3  cos9  'a:  -f  1"=  n* ;' since  Sin" -f  cos2  =•!. 

/.  cos  Z  = 

If  1.33  and  1.55,  the  values  of  n  for  extreme  red  and  violet,  be 
used  in  this  formula,  we  obtain  x,  and  therefore  y  and  z,  for  the 
limiting  angles  of  the  primary  bow. 


RADIUS    OF    RAINBOW    AND    HALO. 


495 


15.  The  Secondary  Bow.— TG  find  the  angle  of  minimum 
deviation.  Using  the  same  notation  as  before,  we  have  in  the 
pentagon  G  ED  BK  (Fig.  8),  G  =  B  — 

180  +  2x  —  6y:  -  a 


_ 

"  dx 

6  cos  x 


=  2- 


=  0. 


=  2 ;  arui  3  cos  x  =  n  cos  y : 
ncos  y 

.*.  9  cos1  x  =  w2  co*2  y ; 


but     sin3  x  =  n9  sin 


/.  8  cos*  x  +  1  = 


.*.  cos 


4  A"  - 1 
*  =  V-$-i 


which,  as  before,  will  furnish  z  for  each  limiting  color  of  the  sec- 
ondary bow. 

16.  The  Common  Halo.  —  Let  D  E  (Fig.  9)  be  the  ray  from 
the  sun,  and  F  G  the  emergent  ray.     Let  D  Ep  —  x\  KE  F  —  y\ 


z  =  x  —  y  4-  y'  —  x'.      Now, 


FIG.  9. 


sin  a:  =  n  sm  y, 
and  sin  y'=  n  sin  #'; 

/.  x  =  sin"1  (ft  sin  ?/), 
and       y'  =  sin"1  (n  sin  #')  = 

sin"1  \n  sin  (tf — y)\. 
By  substitution, 

e  =  sin"1  (n  sin  ?/)  -f  sin"1  \n  sin  (6Y  —  ?/)f  —  C,  Therefore  0  is  i 
function  of  y ;  and,  by  differentiating,  we  have 

d_z  _         n  cos  y  n  cos  (C  —  y)         _  - 

<*#~  fl  -  w"  sin2^       Vl  -  w*  sin^'^'yj  ~ 

n*  cos2  y  y?2  cos'J  (C  -  y} 

"  1  —  n2  sin2  y      1  —  ?i"  sin'  (C  —  y) ' 

1  —  sin2  y          1  —  sin''  (C  —  y} 
'  *  1  —  n2  sm2  ^  ~  1  —  n'  sTn*  (C  —  y) y 
.-.  (w9  -  1)  sin2  y=(rf-  1)  sin2  (C  -  y); 
.:y  =  C  —  y,  and  ?/  —  J  (7; 
and  ^'  =  A  (7. 

Hence,  the  minimum  deviation  occurs  when  the  ray  within  the 
crystal  is  equally  inclined  to  the  sides.  Knowing  n,  the  index  of 
refraction  for  ice,  x,  and  its  °"t™1  ff'  ™iT1  ***  obtained,  and  then  2, 
the  deviation  required. 

'  UNIVERSITY 


